{"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"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938."},"identifying_rows":4,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","2","2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","2","2"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["1","0","1"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["1","0","1"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["-2","-2"]]},"deriv_order":0,"shift":3},{"coeff":{"coeffs":[["-2","-2"]]},"deriv_order":1,"shift":3},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":4},{"coeff":{"coeffs":[["1"]]},"deriv_order":1,"shift":4}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1,2,3,4],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"],["1","1"],["2","4","1"],["2","10","7","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-1","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"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"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["1","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-1"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["0","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["-1","-1"]]},"deriv_order":1}],"shift":1},{"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":1}],"shift":2}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","-1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["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 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 4 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 (LiuWangDerivativeLag) 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"},"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 (LiuWangDerivativeLag) 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"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"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":"liu_wang_derivative_lag","statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 1, 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":[],"parent_interlacing_report":{"conclusion":{"first_index":1,"interlacing":"weak","interlacing_from":2,"root_region":{"upper":{"closed":true,"value":"0"}},"simple_roots":true,"statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0] (so its coefficients have one sign); P_n = x^1 Q_n for n >= 2 with Q_n(0) != 0, and for n >= 3 the Q_n have simple roots, Q_(n-1) strictly interlaces Q_n, and P_(n-1) interlaces P_n (Liu--Wang, common root 0); the rule is applied from n = 4 on in the induction (start moved 1 row(s) past the supplied seeds; P_1, ..., P_3 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 2, ..., 2"},"hypotheses":[{"evidence":"checked exactly","name":"seed_roots_negative","statement":"P_2 / x^1 and P_3 / x^1 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 / x^1 = deg P_2 / x^1 + 1, with leading coefficients of one sign","status":{"kind":"verified"}},{"evidence":"Bezout test, exact","name":"seed_strict_interlacing","statement":"P_2 / x^1 strictly interlaces P_3 / x^1 (simple real roots, no common root)","status":{"kind":"verified"}},{"evidence":"checked exactly for P_1, ..., P_1","name":"earlier_seeds","statement":"the seed rows P_1, ..., P_1 are each zero or real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"lambda_n = 1 > 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of lambda_n = 1 at n = 4 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["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) = 1 > 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of A_n(0) = 1 at n = 4 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"A_n(0) > 0 for every n >= 4","status":{"kind":"verified"}},{"evidence":"V_n = 0","name":"derivative_coefficient_nonpositive","statement":"V_n(x) <= 0 for every x < 0 and every n >= 4","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 >= 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 4 + u have one sign)","name":"lag_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"W_n(x) <= 0 for every x < 0 and every n >= 4","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 > 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 4 + u have one sign)","name":"derivative_or_lag_coefficient_negative","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["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.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"lean_replay":"not_attempted","lean_route":"sign","normal_form":"P_n = x^1 Q_n for n >= 2 (x^1 divides P_2 and P_3, and V_n(0) = 0), with Q_n = A_n(x) Q_(n-1) + V_n(x) Q'_(n-1) + W_n(x) Q_(n-2) for n >= 4, A_n(x) = (1) + (1) x, V_n(x) = 0, W_n(x) = (1) x (A_n = a_n + 1 V_n / x for the rule's a_n(x) = (1) + (1) x)","recurrence":{"first_index":1,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"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 rule is not P_(n+k) = c(x) P_n with c free of n","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"no theorem matched (weighted_renewal_window: support must satisfy offset = 1 + ell * t_degree; fixed_delay_predecessor_window: more than one weighted lag)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: c_2(n, x) = gamma(n) w(x) with w neither constant nor a constant times a square (x + v_0)^2","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"induction start moved 1 row(s) past the seeds: P_n = x^1 Q_n for n >= 2 (x^1 divides P_2 and P_3, and V_n(0) = 0), with Q_n = A_n(x) Q_(n-1) + V_n(x) Q'_(n-1) + W_n(x) Q_(n-2) for n >= 4, A_n(x) = (1) + (1) x, V_n(x) = 0, W_n(x) = (1) x (A_n = a_n + 1 V_n / x for the rule's a_n(x) = (1) + (1) x): every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0] (so its coefficients have one sign); P_n = x^1 Q_n for n >= 2 with Q_n(0) != 0, and for n >= 3 the Q_n have simple roots, Q_(n-1) strictly interlaces Q_n, and P_(n-1) interlaces P_n (Liu--Wang, common root 0); the rule is applied from n = 4 on in the induction (start moved 1 row(s) past the supplied seeds; P_1, ..., P_3 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 2, ..., 2","route":"liu_wang_derivative_lag","success":true,"variant":"liu_wang_derivative_lag"}],"conditional_on":[],"interlacing":{"from_index":2,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = x^1 Q_n for n >= 2 (x^1 divides P_2 and P_3, and V_n(0) = 0), with Q_n = A_n(x) Q_(n-1) + V_n(x) Q'_(n-1) + W_n(x) Q_(n-2) for n >= 4, A_n(x) = (1) + (1) x, V_n(x) = 0, W_n(x) = (1) x (A_n = a_n + 1 V_n / x for the rule's a_n(x) = (1) + (1) x)","every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0] (so its coefficients have one sign); P_n = x^1 Q_n for n >= 2 with Q_n(0) != 0, and for n >= 3 the Q_n have simple roots, Q_(n-1) strictly interlaces Q_n, and P_(n-1) interlaces P_n (Liu--Wang, common root 0); the rule is applied from n = 4 on in the induction (start moved 1 row(s) past the supplied seeds; P_1, ..., P_3 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 2, ..., 2"],"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":"weak","orientation":"previous_interlaces_next"},"route":"liu_wang_derivative_lag","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 = phi(n, D) = 1 + (1) D for every n >= 1, the side conditions hold for every n, and every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, 0]"},"kind":"preserver_image"}