{"chains":[{"edge":{"id":"e-4fb0c58de9dbd95f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-35e99788be157971","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"36e2db3d2745b1d4ee84edbaa3ddf01091b9cabb25c3e44fd2122817235427be","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0b318b4ec2091cb2","role":"sequence","name":"Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=n.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"2"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 2)"}]},"premises":[{"premise":{"position":0,"node_id":"n-0b318b4ec2091cb2","role":"sequence","name":"Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=n.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"2"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 2)"},"chains":[{"edge":{"id":"e-d92e4a9b9719ac57","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-0b318b4ec2091cb2","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"7d45f4dffa8a0c85bfc32b02895f2b8bbd5e5557a1db376aa3f926048002991e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"2"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}]}]}],"truncated":false,"expanded":2}