{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-6"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"derivative of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1)."},"identifying_rows":4,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","-2"],["-1","7"],["-1","-3"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","1","-1"],["0","-2","2"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["2"],["-3"],["1"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1,2,3],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["0"],["0"],["0"],["6"]],"recurrence":{"denominator":{"coeffs":[["6"],["-5"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","12"],["-1","-13"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-3","3"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":3,"multiplier":["-1","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-6"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":2,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","-2","2"],["0","3","-3"],["0","-1","1"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["2"],["-3"],["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["-2","4"],["0","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","-1","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","3","-3"],["0","-3","3"]]},"deriv_order":0}],"row_multiplier":["0","-1","1"],"start_base":3,"step_multiplier":{"coeffs":[["0","0","1","-2","1"]]},"step_quotient":[{"coeff":{"coeffs":[["0","0","3","-9","9","-3"],["0","0","1","1","-5","3"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","2","-6","6","-2"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","3"],["-1","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-1"],["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 2) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = phi(n, D) = (1) D; so the candidate rows satisfy the input rule at every base n >= 3 (the exceptional bases checked exactly), and agreement on the 4 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 parent theorem does not cover every parent row used","status":"none"},"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 [0, 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 [0, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 0 + 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":"0","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":0,"input_interlacing":{"reason":"the parent theorem does not cover every parent row used","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"phi(n, D) = (1) D","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-6"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"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) D, for every n >= 0"},"parent_proof":{"route":"liu_wang_derivative_lag","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":"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":["the division remainder is removed by row annihilators of the shifted parent"],"parent_interlacing_report":{"conclusion":{"first_index":2,"interlacing":"strict","interlacing_from":3,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":true,"statement":"P_0, P_1 are zero; every other P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 3, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 4; the rule is applied from n = 4 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_3 enter as seed rows, checked exactly)"},"hypotheses":[{"evidence":"f(n) = n - 2 != 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of f(n) = n - 2 at n = 4 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["2","1"]},"numerator":["-2","1"],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 4","status":{"kind":"verified"}},{"evidence":"checked exactly","name":"seed_roots_negative","statement":"P_2 and P_3 are nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"degrees 0 and 1, leading coefficients of one sign","name":"seed_degree_step","statement":"deg P_3 = deg P_2 + 1, with leading coefficients of one sign","status":{"kind":"verified"}},{"evidence":"Bezout test, exact","name":"seed_strict_interlacing","statement":"P_2 strictly interlaces P_3 (simple real roots, no common root)","status":{"kind":"verified"}},{"evidence":"checked exactly for P_0, ..., P_1 (2 zero)","name":"earlier_seeds","statement":"the seed rows P_0, ..., P_1 are each zero or real-rooted with every root in (-inf, 0)","status":{"kind":"verified"}},{"evidence":"lambda_n = (n)/(n - 2) > 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of lambda_n = (n)/(n - 2) at n = 4 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["2","1"],"shifted_numerator":["4","1"]},"numerator":["0","1"],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] A_n + (deg P_(n-1)) [x^2] V_n > 0 for every n >= 4 (deg P_n = deg P_(n-1) + 1 with leading coefficients of one sign)","status":{"kind":"verified"}},{"evidence":"[x^0] c_1 and [x^0] c_2 are zero polynomials","name":"multipliers_vanish_at_zero","statement":"V_n(0) = 0 and W_n(0) = 0 for every n >= 4","status":{"kind":"verified"}},{"evidence":"A_n(0) = (n)/(n - 2) > 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of A_n(0) = (n)/(n - 2) at n = 4 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["2","1"],"shifted_numerator":["4","1"]},"numerator":["0","1"],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"A_n(0) > 0 for every n >= 4","status":{"kind":"verified"}},{"evidence":"V_n(x) = gamma(n) w(x) / f(n) with w(x) = -2 x^2 + 2 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (1)/(n - 2) >= 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (1)/(n - 2) at n = 4 + u have one sign)","name":"derivative_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["2","1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"V_n(x) <= 0 for every x < 0 and every n >= 4","status":{"kind":"verified"}},{"evidence":"W_n = 0","name":"lag_coefficient_nonpositive","statement":"W_n(x) <= 0 for every x < 0 and every n >= 4","status":{"kind":"verified"}},{"evidence":"V_n(x) = gamma(n) w(x) / f(n) with w(x) = -2 x^2 + 2 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (1)/(n - 2) > 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (1)/(n - 2) at n = 4 + u have one sign)","name":"derivative_or_lag_coefficient_negative","sign_decision":{"certificate":{"condition":"positive","denominator":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["2","1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"for every n >= 4 and x < 0, V_n(x) < 0 or W_n(x) < 0 (V_n < 0 on (-inf, 0) for every n, or W_n < 0 there for every n)","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.derivLag_strictInterl_of_nonnegCoeffs","RealRooted.derivLag_strictInterl_and_noCommonRoot","RealRooted.derivLag_eval_zero_pos"],"lean_replay":"not_attempted","lean_route":"halfGrowth","normal_form":"P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 4, with A_n(x) = ((n)/(n - 2)) + ((3 n - 6)/(n - 2)) x, V_n(x) = ((2)/(n - 2)) x + ((-2)/(n - 2)) x^2, W_n(x) = 0","recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-6"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","schema":"polytool.recurrence-real-rootedness-theorem.v1","start_delay":1,"status":"proved","theorem":"liu_wang_derivative_lag"},"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":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 3, with a_n(x) = ((n)/(n - 2)) + ((3 n - 6)/(n - 2)) x and b_n(x) = ((0)/(n - 2)) + ((2)/(n - 2)) x + ((-2)/(n - 2)) x^2; open hypotheses: earlier_seeds: P_0 is zero or not real-rooted with every root in (-inf, 0) (exact)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"induction start moved 1 row(s) past the seeds: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 4, with A_n(x) = ((n)/(n - 2)) + ((3 n - 6)/(n - 2)) x, V_n(x) = ((2)/(n - 2)) x + ((-2)/(n - 2)) x^2, W_n(x) = 0: P_0, P_1 are zero; every other P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 3, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 4; the rule is applied from n = 4 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_3 enter as seed rows, checked exactly)","route":"liu_wang_derivative_lag","success":true,"variant":"liu_wang_derivative_lag"}],"conditional_on":[],"interlacing":{"from_index":3,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 4, with A_n(x) = ((n)/(n - 2)) + ((3 n - 6)/(n - 2)) x, V_n(x) = ((2)/(n - 2)) x + ((-2)/(n - 2)) x^2, W_n(x) = 0","P_0, P_1 are zero; every other P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 3, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 4; the rule is applied from n = 4 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_3 enter as seed rows, checked exactly)"],"theorem":"liu_wang_derivative_lag: L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"liu_wang_derivative_lag","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 = phi(n, D) = (1) D 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"}