{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["6","3"]],"recurrence":{"denominator":{"coeffs":[["0"],["6"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","-2"],["0","0","-3"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["20","10"],["18","9"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"existing definition"},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-42","-21"],["-26","-16"],["-4","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","6","6"],["0","2","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["7"],["8"],["1"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["6"],["1"]]},"terms":[{"coeff":{"coeffs":[["20","8"],["18","10"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-4"],["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["7","8","1"],"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["6","3"]],"recurrence":{"denominator":{"coeffs":[["0"],["6"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","-2"],["0","0","-3"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["20","10"],["18","9"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-42","-21"],["-26","-16"],["-4","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","6","6"],["0","2","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["7"],["8"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["7"],["8"],["1"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["0","49","49"],["0","112","112"],["0","78","78"],["0","16","16"],["0","1","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","-448","-448"],["0","0","-1352","-1352"],["0","0","-1514","-1514"],["0","0","-785","-785"],["0","0","-197","-197"],["0","0","-23","-23"],["0","0","-1","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","896","1792","896"],["0","0","1808","3616","1808"],["0","0","1220","2440","1220"],["0","0","350","700","350"],["0","0","44","88","44"],["0","0","2","4","2"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","-5376","-10752","-5376"],["0","0","-12640","-25280","-12640"],["0","0","-10936","-21872","-10936"],["0","0","-4540","-9080","-4540"],["0","0","-964","-1928","-964"],["0","0","-100","-200","-100"],["0","0","-4","-8","-4"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["3"],["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-6","-2"],["-2","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":2,"shift":0}]},"quotient":[],"step_multiplier":{"coeffs":[["0","0","343","686","343"],["0","0","1176","2352","1176"],["0","0","1491","2982","1491"],["0","0","848","1696","848"],["0","0","213","426","213"],["0","0","24","48","24"],["0","0","1","2","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","3136","6272","3136"],["0","0","13048","26096","13048"],["0","0","21862","43724","21862"],["0","0","18959","37918","18959"],["0","0","9173","18346","9173"],["0","0","2522","5044","2522"],["0","0","388","776","388"],["0","0","31","62","31"],["0","0","1","2","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","-6272","-15680","-12544","-3136"],["0","0","-19824","-50344","-41216","-10696"],["0","0","-23900","-62032","-52364","-14232"],["0","0","-14018","-37462","-32870","-9426"],["0","0","-4328","-11968","-10952","-3312"],["0","0","-716","-2044","-1940","-612"],["0","0","-60","-176","-172","-56"],["0","0","-2","-6","-6","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","784","2352","2352","784"],["0","0","0","2282","6846","6846","2282"],["0","0","0","2417","7251","7251","2417"],["0","0","0","1148","3444","3444","1148"],["0","0","0","254","762","762","254"],["0","0","0","26","78","78","26"],["0","0","0","1","3","3","1"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","37632","91728","70560","16464"],["0","0","131488","323442","252420","60466"],["0","0","183048","455717","362290","89621"],["0","0","131908","333442","271160","69626"],["0","0","54004","138961","115910","30953"],["0","0","12952","33958","29060","8054"],["0","0","1792","4783","4190","1199"],["0","0","132","358","320","94"],["0","0","4","11","10","3"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-4704","-14112","-14112","-4704"],["0","0","0","-15260","-45780","-45780","-15260"],["0","0","0","-19066","-57198","-57198","-19066"],["0","0","0","-11722","-35166","-35166","-11722"],["0","0","0","-3820","-11460","-11460","-3820"],["0","0","0","-664","-1992","-1992","-664"],["0","0","0","-58","-174","-174","-58"],["0","0","0","-2","-6","-6","-2"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","12"],["0","0","7"],["0","0","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-72","-36"],["-34","-17"],["-4","-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["16"],["10"],["1"]]},"deriv_order":0,"shift":2}]},"statement":"w(n) E = sum_nu H_nu S^nu L' (with any remainder removed by row annihilators of the parent), where L' is the parent rule at P'_n = P_(n+0) (valid from base 0) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = P((1) x + (0)); so the candidate rows satisfy the input rule at every base n >= 0 (the exceptional bases checked exactly), and agreement on the 1 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"from_index":1,"justification":"the parent theorem (ThreeTermRecurrence) proves P_(m-1) strictly interlacing P_m for every m >= 1; T = P((1) x + (0)) is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 1","kind":"weak","orientation":"unordered","status":"claimed"},"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"lift_summary":{"hypotheses":[{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":0,"input_interlacing":{"interlacing":{"from_index":1,"justification":{"premises":["the parent theorem (ThreeTermRecurrence) proves P_(m-1) strictly interlacing P_m for every m >= 1; T = P((1) x + (0)) is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 1"],"theorem":"Hermite--Kakeya--Obreschkoff theorem (N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, 1963; see S. Fisk, Polynomials, Roots, and Interlacing, arXiv:math/0612833): real polynomials f, g have weakly interlacing zeros iff alpha f + beta g is real-rooted or zero for all real alpha, beta; a linear T independent of n that maps every real-rooted polynomial to a real-rooted polynomial or zero therefore maps interlacing pairs to pairs with weakly interlacing zeros (orientation not determined)"},"kind":"weak","orientation":"unordered"},"status":"proved"},"lean_replay":"not_attempted","operation":{"description":"P((1) x + (0))","kind":"preserver_image","operator":{"intercept":"0","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","3"]],"recurrence":{"denominator":{"coeffs":[["0"],["6"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","-2"],["0","0","-3"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["20","10"],["18","9"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (0)), for every n >= 0"},"parent_proof":{"route":"three_term_recurrence","statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]","input_root_region":{"upper":{"closed":true,"value":"0"}},"parent_hypothesis":"every parent row used is zero or real-rooted with all roots in (-inf, 0] (coefficients of one sign)","parent_root_region":{"upper":{"closed":true,"value":"0"}},"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"},"verdict":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_interlacing_report":{"conclusion":{"first_index":0,"interlacing":"strict","interlacing_from":1,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":true,"statement":"every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)"},"hypotheses":[{"evidence":"f(n) = n^2 + 6 n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = n^2 + 6 n at n = 2 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["16","10","1"]},"numerator":["0","6","1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 2","status":{"kind":"verified"}},{"evidence":"f(n) a_n = 4 n^2 + 18 n + 20 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 4 n^2 + 18 n + 20 at n = 2 + u have one sign)","name":"leading_coefficient_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["72","34","4"]},"numerator":["20","18","4"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"a_n != 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"Q_0 has degree 0, Q_1 degree 1; simple real roots and strict interlacing checked exactly","name":"seeds_interlace","statement":"Q_0 and Q_1 have simple real roots, degrees e and e + 1, and Q_0 strictly interlaces Q_1","status":{"kind":"verified"}},{"evidence":"beta_2 = 1/36 (exact)","name":"first_coefficient_positive","statement":"beta_2 = c_2 L_0 / (a_2 L_1) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = (1/16 n^4 + 7/16 n^3 + 9/16 n^2 - 7/16 n - 5/8)/(n^4 + 7 n^3 + 71/4 n^2 + 77/4 n + 15/2) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^4 + 7/16 n^3 + 9/16 n^2 - 7/16 n - 5/8)/(n^4 + 7 n^3 + 71/4 n^2 + 77/4 n + 15/2) at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["15/2","77/4","71/4","7","1"],"method":{"kind":"taylor_shift","shifted_denominator":["495","1691/4","539/4","19","1"],"shifted_numerator":["20","43/2","63/8","19/16","1/16"]},"numerator":["-5/8","-7/16","9/16","7/16","1/16"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"beta_n = c_n / (a_n a_(n-1)) > 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"seeds checked exactly; positive-ratio induction at t = 0 with g(n) = (1/4 n + 3/2)/(n + 5/2) (the exact ratio r_n): g(n) > 0 for n >= 2, r_2 >= g(2) exactly, and (t - alpha_n) - beta_n/g(n-1) - g(n) >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of (t - alpha_n) - beta_n/g(n-1) - g(n) at n = 3 + u have one sign)","name":"root_upper_bound","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1875/4","7625/4","52775/16","25425/8","29871/16","2739/4","153","19","1"],"method":{"kind":"taylor_shift","shifted_denominator":["598950","3695945/4","9933803/16","1898415/8","902691/16","34167/4","804","43","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"every root of Q_n is < 0 for every n >= 0 (region (-inf, 0) for P)","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 2, with a_n = (4 n^2 + 18 n + 20)/(n^2 + 6 n), b_n = (2 n^2 + 9 n + 10)/(n^2 + 6 n), c_n = (n^2 + 3 n + 2)/(n^2 + 6 n); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2)","recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","3"]],"recurrence":{"denominator":{"coeffs":[["0"],["6"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","-2"],["0","0","-3"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["20","10"],["18","9"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"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","requested_root_region":{"upper":{"closed":false,"value":"0"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"},"parent_proof":{"attempts":[{"detail":"the leader is not a nonzero constant","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"no theorem matched (weighted_renewal_window: LHS must be a nonzero constant; fixed_delay_predecessor_window: LHS must be a nonzero constant)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 2, with a_n = (4 n^2 + 18 n + 20)/(n^2 + 6 n), b_n = (2 n^2 + 9 n + 10)/(n^2 + 6 n), c_n = (n^2 + 3 n + 2)/(n^2 + 6 n); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2): every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)","route":"three_term_recurrence","success":true,"variant":"three_term_recurrence"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 2, with a_n = (4 n^2 + 18 n + 20)/(n^2 + 6 n), b_n = (2 n^2 + 9 n + 10)/(n^2 + 6 n), c_n = (n^2 + 3 n + 2)/(n^2 + 6 n); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2)","every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)"],"theorem":"three_term_recurrence: 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"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"three_term_recurrence","rows_checked":10,"statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (0)) for every n >= 0, the side conditions hold for every n, and every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]"},"kind":"definition_equality"}