{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"-6","shape":"affine","slope":"1"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["5","1"],["-2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","6","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1. shifted by x ->  x + 6"},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-1","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","-1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1,2],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"],["-5","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-5","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","1","1"],"operator":{"intercept":"-6","shape":"affine","slope":"1"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["5","1"],["-2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","6","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","0","0","0"],["-15","-38","-32","-10","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["25","85","106","58","13","1"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["0","0","0","0"],["5","11","7","1"]]},"deriv_order":0}],"shift":1}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["1","1"],["-2","-1"],["1","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["9","12","3"],["-5","-6","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["5","11","7","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["25","60","46","12","1"]]},"deriv_order":0}],"row_multiplier":["5","11","7","1"],"start_base":1,"step_multiplier":{"coeffs":[["25","110","191","164","71","14","1"]]},"step_quotient":[{"coeff":{"coeffs":[["-25","-85","-81","27","93","57","13","1"],["-50","-220","-382","-328","-142","-28","-2","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["125","700","1640","2076","1530","660","160","20","1"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-3","-1"],["2","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-5","-6","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0"],["1"]]},"deriv_order":0,"shift":1}]},"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 1) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = P((1) x + (-6)); so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 2 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"interlacing":{"from_index":2,"justification":"the parent theorem (MaWangInterval) proves P_(m-1) weakly interlacing P_m for every m >= 2; T = P((1) x + (-6)) 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 >= 2","kind":"weak","orientation":"unordered","status":"claimed"},"interval_parent_proof":{"conclusion":{"first_index":1,"interlacing":"weak","interlacing_from":2,"root_region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":true,"value":"-1"}},"simple_roots":false,"statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in (-5, -1]; deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 2"},"hypotheses":[{"evidence":"f(n) = n - 1 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = n - 1 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":["1","1"]},"numerator":["-1","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":"P_1 has degree 0, the root -5 with multiplicity 0 and the root -1 with multiplicity 0; checked exactly","name":"seed_roots_in_interval","statement":"P_1 is nonzero and real-rooted with every root in [-5, -1]","status":{"kind":"verified"}},{"evidence":"a_n = 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of a_n = 1 at n = 2 + u have one sign)","name":"slope_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"a_n = [x^1] c_0(n, x) / gamma(n) > 0 for every n >= 2 (the coefficient a of the normalized step, so deg P_n = deg P_(n-1) + 1)","status":{"kind":"verified"}},{"evidence":"no hit before n = 2; lattice argument from n = 2 (multiplicity 0): e(n) = -1/2 n + 1 is affine with slope -1/2 and 1/1/2 = 2 is an integer, and E(2) = 0 is a nonnegative multiple of 1/2; E drops by 1/2 per step and returns from 0 to 1/2 after each hit, so E(n') >= 0 for every n' >= 2, with a hit every 2 steps","name":"upper_endpoint_threshold","statement":"E(n) = e(n) + m(n) >= 0 for every n >= 2, with e(n) = -1/2 n + 1 at the end U = -1, m(2) = 0 the multiplicity of U = -1 in the last seed, and m raised by one after every n with E(n) = 0","status":{"kind":"verified"}},{"evidence":"no hit before n = 2; from n = 2 on (multiplicity 0): e(n) + 0 = 1/2 n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of e(n) + 0 = 1/2 n at n = 2 + u have one sign)","name":"lower_endpoint_threshold","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1","1/2"]},"numerator":["0","1/2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"E(n) = e(n) + m(n) >= 0 for every n >= 2, with e(n) = 1/2 n at the end L = -5, m(2) = 0 the multiplicity of L = -5 in the last seed, and m raised by one after every n with E(n) = 0","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.affineEndpointStep_closed","RealRooted.affineEndpointStep_roots_mem_Icc_iff","RealRooted.affineEndpointStep_rootMultiplicity_zero","RealRooted.affineEndpointStep_rootMultiplicity_neg_one","RealRooted.MaWang.endpointStep","RealRooted.MaWang.affineEndpointSequence","RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff"],"lean_replay":"not_attempted","normal_form":"P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 2, with L = -5, U = -1, alpha_n = (1)/(n - 1), beta_n = (-2 n + 5)/(n - 1), kappa_n = (1)/(n - 1)","recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["5","1"],["-2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","6","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"ma_wang_interval"},"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":1,"input_interlacing":{"interlacing":{"from_index":2,"justification":{"premises":["the parent theorem (MaWangInterval) proves P_(m-1) weakly interlacing P_m for every m >= 2; T = P((1) x + (-6)) 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 >= 2"],"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 + (-6))","kind":"preserver_image","operator":{"intercept":"-6","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["5","1"],["-2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","6","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (-6)), for every n >= 1"},"parent_proof":{"route":"ma_wang_interval","statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in (-5, -1]; deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 2","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 1, is zero or real-rooted with roots in (1, 5]","input_root_region":{"lower":{"closed":false,"value":"1"},"upper":{"closed":true,"value":"5"}},"parent_hypothesis":"every parent row used is real-rooted with every root in (-5, -1]","parent_root_region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":true,"value":"-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"},"verdict":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_interlacing_report":{"conclusion":{"first_index":1,"interlacing":"weak","interlacing_from":2,"root_region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":true,"value":"-1"}},"simple_roots":false,"statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in (-5, -1]; deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 2"},"hypotheses":[{"evidence":"f(n) = n - 1 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = n - 1 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":["1","1"]},"numerator":["-1","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":"P_1 has degree 0, the root -5 with multiplicity 0 and the root -1 with multiplicity 0; checked exactly","name":"seed_roots_in_interval","statement":"P_1 is nonzero and real-rooted with every root in [-5, -1]","status":{"kind":"verified"}},{"evidence":"a_n = 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of a_n = 1 at n = 2 + u have one sign)","name":"slope_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"a_n = [x^1] c_0(n, x) / gamma(n) > 0 for every n >= 2 (the coefficient a of the normalized step, so deg P_n = deg P_(n-1) + 1)","status":{"kind":"verified"}},{"evidence":"no hit before n = 2; lattice argument from n = 2 (multiplicity 0): e(n) = -1/2 n + 1 is affine with slope -1/2 and 1/1/2 = 2 is an integer, and E(2) = 0 is a nonnegative multiple of 1/2; E drops by 1/2 per step and returns from 0 to 1/2 after each hit, so E(n') >= 0 for every n' >= 2, with a hit every 2 steps","name":"upper_endpoint_threshold","statement":"E(n) = e(n) + m(n) >= 0 for every n >= 2, with e(n) = -1/2 n + 1 at the end U = -1, m(2) = 0 the multiplicity of U = -1 in the last seed, and m raised by one after every n with E(n) = 0","status":{"kind":"verified"}},{"evidence":"no hit before n = 2; from n = 2 on (multiplicity 0): e(n) + 0 = 1/2 n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of e(n) + 0 = 1/2 n at n = 2 + u have one sign)","name":"lower_endpoint_threshold","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1","1/2"]},"numerator":["0","1/2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"E(n) = e(n) + m(n) >= 0 for every n >= 2, with e(n) = 1/2 n at the end L = -5, m(2) = 0 the multiplicity of L = -5 in the last seed, and m raised by one after every n with E(n) = 0","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.affineEndpointStep_closed","RealRooted.affineEndpointStep_roots_mem_Icc_iff","RealRooted.affineEndpointStep_rootMultiplicity_zero","RealRooted.affineEndpointStep_rootMultiplicity_neg_one","RealRooted.MaWang.endpointStep","RealRooted.MaWang.affineEndpointSequence","RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff"],"lean_replay":"not_attempted","normal_form":"P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 2, with L = -5, U = -1, alpha_n = (1)/(n - 1), beta_n = (-2 n + 5)/(n - 1), kappa_n = (1)/(n - 1)","recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["5","1"],["-2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","6","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"ma_wang_interval"},"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":"not applicable: the theorem needs at least two seed rows","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 2, with a_n(x) = ((-2 n + 5)/(n - 1)) + ((1)/(n - 1)) x and b_n(x) = ((5)/(n - 1)) + ((6)/(n - 1)) x + ((1)/(n - 1)) x^2; open hypotheses: derivative_coefficient_vanishes_at_zero: [x^0] c_1(n, x) = 5 is not the zero polynomial; derivative_coefficient_nonpositive: b_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] b_n <= 0 fails for k = 0 (not a disproof): fails at n = 2: [x^0] b_n = 5 there (exact); constant_term_positive: fails at n = 3: a_n(0) = (-2 n + 5)/(n - 1) = -1/2 there (exact)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"not applicable: the theorem needs two seed rows","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 2, with L = -5, U = -1, alpha_n = (1)/(n - 1), beta_n = (-2 n + 5)/(n - 1), kappa_n = (1)/(n - 1): every P_n, n >= 1, is nonzero and real-rooted with every root in (-5, -1]; deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 2","route":"ma_wang_interval","success":true,"variant":"ma_wang_interval"}],"conditional_on":[],"interlacing":{"from_index":2,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 2, with L = -5, U = -1, alpha_n = (1)/(n - 1), beta_n = (-2 n + 5)/(n - 1), kappa_n = (1)/(n - 1)","every P_n, n >= 1, is nonzero and real-rooted with every root in (-5, -1]; deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 2"],"theorem":"ma_wang_interval: S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction"},"kind":"weak","orientation":"previous_interlaces_next"},"route":"ma_wang_interval","rows_checked":10,"statement":"every parent row P_n, n >= 1, 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 + (-6)) for every n >= 1, the side conditions hold for every n, and every P_n, n >= 1, is nonzero and real-rooted with every root in (-5, -1]; deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 2; so every Q_n, n >= 1, is zero or real-rooted with roots in (1, 5]"},"kind":"preserver_image"}