{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x))."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["2","-5","4","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["2","-4","1","2","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","1","-3","3","-1"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-3","5","-1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["-3","4"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","-1","1"]]},"deriv_order":2,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["1"]]},"deriv_order":1,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1,2],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"],["0","-1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2","5","-4","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1","3","-3","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","-5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-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":[["-3","4","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["-1","2","-1"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":1}],"shift":1}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-1","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["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 = phi(n, D) = 1 + (1) D; 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 = phi(n, D) = 1 + (1) D 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":"0"},"upper":{"closed":true,"value":"1"}},"simple_roots":false,"statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in (0, 1]; deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 2"},"hypotheses":[{"evidence":"P_1 has degree 0, the root 0 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 [0, 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; from n = 2 on (multiplicity 0): e(n) + 0 = 0 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of e(n) + 0 = 0 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":[]},"numerator":[],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"E(n) = e(n) + m(n) >= 0 for every n >= 2, with e(n) = 0 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 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of e(n) + 0 = 1 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"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"E(n) = e(n) + m(n) >= 0 for every n >= 2, with e(n) = 1 at the end L = 0, m(2) = 0 the multiplicity of L = 0 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 = 0, U = 1, alpha_n = -1, beta_n = 1, kappa_n = -1","recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-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.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"derivative_polynomial_real_rooted","statement":"the polynomial in D is real-rooted for every n >= n0","status":{"kind":"verified"}},{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}},{"evidence":"delta(n) != 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 1 + u have one sign)","name":"operator_coefficients_defined","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"the operator coefficients are finite at every n >= n0 (their denominators do not vanish)","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 = phi(n, D) = 1 + (1) D 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":"phi(n, D) = 1 + (1) D","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D, 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 (0, 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 (-inf, +inf)","input_root_region":{},"parent_hypothesis":"every parent row used is real-rooted with every root in (0, 1]","parent_root_region":{"lower":{"closed":false,"value":"0"},"upper":{"closed":true,"value":"1"}},"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n"},"verdict":"proved"},"notes":[],"parent_interlacing_report":{"conclusion":{"first_index":1,"interlacing":"weak","interlacing_from":2,"root_region":{"lower":{"closed":false,"value":"0"},"upper":{"closed":true,"value":"1"}},"simple_roots":false,"statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in (0, 1]; deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 2"},"hypotheses":[{"evidence":"P_1 has degree 0, the root 0 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 [0, 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; from n = 2 on (multiplicity 0): e(n) + 0 = 0 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of e(n) + 0 = 0 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":[]},"numerator":[],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"E(n) = e(n) + m(n) >= 0 for every n >= 2, with e(n) = 0 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 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of e(n) + 0 = 1 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"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"E(n) = e(n) + m(n) >= 0 for every n >= 2, with e(n) = 1 at the end L = 0, m(2) = 0 the multiplicity of L = 0 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 = 0, U = 1, alpha_n = -1, beta_n = 1, kappa_n = -1","recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-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":[],"conditional_on":[],"failing_row_index":2,"rows_checked":10,"statement":"parent row P_2 is nonzero and not real-rooted with coefficients of one sign; this parent gives no proof","status":"refuted"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D 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 (0, 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 (-inf, +inf)"},"kind":"preserver_image"}