{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"shift_x_plus_one of chebyshev_t"},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["-1","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0"],["1"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-3"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","1"],"operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","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"],["1"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["-1","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2","-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 = x^(d_n) P(1/x), d_n = 0 + 1 (n - 0); 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 >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"reason":"the schedule d_n depends on n","status":"none"},"interval_parent_proof":{"conclusion":{"first_index":0,"interlacing":"weak","interlacing_from":2,"root_region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 0, is nonzero and real-rooted with every root in (-2, 0); 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 1, the root -2 with multiplicity 0 and the root 0 with multiplicity 0; checked exactly","name":"seed_roots_in_interval","statement":"P_1 is nonzero and real-rooted with every root in [-2, 0]","status":{"kind":"verified"}},{"evidence":"checked exactly for P_0, ..., P_0","name":"earlier_seeds","statement":"the seed rows P_0, ..., P_0 are nonzero and real-rooted with every root in [-2, 0]","status":{"kind":"verified"}},{"evidence":"a_n = n - 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of a_n = 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","1"]},"numerator":["-1","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; from n = 2 on (multiplicity 0): e(n) + 0 = 1/2 n - 1/2 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of e(n) + 0 = 1/2 n - 1/2 at n = 2 + u have one sign)","name":"upper_endpoint_threshold","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1/2","1/2"]},"numerator":["-1/2","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 - 1/2 at the end U = 0, m(2) = 0 the multiplicity of U = 0 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 - 1/2 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of e(n) + 0 = 1/2 n - 1/2 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":["1/2","1/2"]},"numerator":["-1/2","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 - 1/2 at the end L = -2, m(2) = 0 the multiplicity of L = -2 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 = -2, U = 0, alpha_n = (n - 1)/(n - 1), beta_n = (n - 1)/(n - 1), kappa_n = (1)/(n - 1)","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","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":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"lean_route":"reversed","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":{"reason":"the schedule d_n depends on n","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"x^(d_n) P(1/x), d_n = 0 + 1 (n - 0)","kind":"preserver_image","operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = x^(d_n) P(1/x), d_n = 0 + 1 (n - 0), for every n >= 0"},"parent_proof":{"route":"ma_wang_interval","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":[],"parent_interlacing_report":{"conclusion":{"first_index":0,"interlacing":"weak","interlacing_from":2,"root_region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 0, is nonzero and real-rooted with every root in (-2, 0); 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 1, the root -2 with multiplicity 0 and the root 0 with multiplicity 0; checked exactly","name":"seed_roots_in_interval","statement":"P_1 is nonzero and real-rooted with every root in [-2, 0]","status":{"kind":"verified"}},{"evidence":"checked exactly for P_0, ..., P_0","name":"earlier_seeds","statement":"the seed rows P_0, ..., P_0 are nonzero and real-rooted with every root in [-2, 0]","status":{"kind":"verified"}},{"evidence":"a_n = n - 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of a_n = 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","1"]},"numerator":["-1","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; from n = 2 on (multiplicity 0): e(n) + 0 = 1/2 n - 1/2 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of e(n) + 0 = 1/2 n - 1/2 at n = 2 + u have one sign)","name":"upper_endpoint_threshold","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1/2","1/2"]},"numerator":["-1/2","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 - 1/2 at the end U = 0, m(2) = 0 the multiplicity of U = 0 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 - 1/2 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of e(n) + 0 = 1/2 n - 1/2 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":["1/2","1/2"]},"numerator":["-1/2","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 - 1/2 at the end L = -2, m(2) = 0 the multiplicity of L = -2 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 = -2, U = 0, alpha_n = (n - 1)/(n - 1), beta_n = (n - 1)/(n - 1), kappa_n = (1)/(n - 1)","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","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":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"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 rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_2(n, x) P_(n-2) with both terms nonzero","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"Unresolved: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 2, with a_n(x) = ((n - 1)/(n - 1)) + ((n - 1)/(n - 1)) x and b_n(x) = ((0)/(n - 1)) + ((2)/(n - 1)) x + ((1)/(n - 1)) x^2; open hypotheses: 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 = 2 (not a disproof): fails at n = 2: [x^2] b_n = 1 there (exact)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"Unresolved: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = ((n - 1)/(n - 1)) + ((n - 1)/(n - 1)) x, V_n(x) = ((2)/(n - 1)) x + ((1)/(n - 1)) x^2, W_n(x) = 0; open hypotheses: derivative_coefficient_nonpositive: V_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] V_n <= 0 fails for k = 2 (not a disproof): fails at n = 2: [x^2] V_n = 1 there (exact); derivative_or_lag_coefficient_negative: V_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] V_n <= 0 fails for k = 2 (not a disproof): fails at n = 2: [x^2] V_n = 1 there (exact)","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 = -2, U = 0, alpha_n = (n - 1)/(n - 1), beta_n = (n - 1)/(n - 1), kappa_n = (1)/(n - 1): every P_n, n >= 0, is nonzero and real-rooted with every root in (-2, 0); 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 = -2, U = 0, alpha_n = (n - 1)/(n - 1), beta_n = (n - 1)/(n - 1), kappa_n = (1)/(n - 1)","every P_n, n >= 0, is nonzero and real-rooted with every root in (-2, 0); 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 >= 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 = x^(d_n) P(1/x), d_n = 0 + 1 (n - 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":"preserver_image"}