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],["0","0","0","0","0","336","2688","5376","0"],["0","0","0","0","0","0","96","768","1536"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","-13","-126","-384","-352"],["0","0","0","0","0","0","48","464","1408","1280"],["0","0","0","0","0","0","24","216","576","384"],["0","0","0","0","0","0","-256","-2352","-6528","-4864"],["0","0","0","0","0","0","240","2048","4864","2048"],["0","0","0","0","0","0","0","192","1536","3072"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","0","-1","-8","-16"],["0","0","0","0","6","48","96"],["0","0","0","0","-12","-96","-192"],["0","0","0","0","8","64","128"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","2","18","48","32"],["0","0","0","0","-6","-50","-112","-32"],["0","0","0","0","0","-12","-96","-192"],["0","0","0","0","8","72","192","128"],["0","0","0","0","0","16","128","256"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","-2","-18","-48","-32"],["0","0","0","0","0","6","48","96","0"],["0","0","0","0","0","0","24","192","384"],["0","0","0","0","0","-8","-96","-384","-512"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","1","12","48","64"],["0","0","0","0","0","0","-4","-48","-192","-256"],["0","0","0","0","0","0","4","48","192","256"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","0","2","19","56","48"],["0","0","0","0","-6","-54","-144","-96"],["0","0","0","0","0","-12","-96","-192"],["0","0","0","0","8","88","320","384"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","-1","-12","-48","-64"],["0","0","0","0","0","4","48","192","256"],["0","0","0","0","0","-4","-48","-192","-256"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","0","0","8","64","128"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","-2"]]},"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 0) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = f(n, theta) = 1 + (2) theta + (1) theta^2; so the candidate rows satisfy the input rule at every base n >= 0 (the exceptional bases checked exactly), and agreement on the 1 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":{"from_index":1,"justification":"the parent theorem (LiuWangDerivativeLag) proves P_(m-1) weakly interlacing P_m for every m >= 1; T = f(n, theta) = 1 + (2) theta + (1) theta^2 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 >= 1","kind":"weak","orientation":"unordered","status":"claimed"},"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 2, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"euler_polynomial_roots_nonpositive","statement":"the polynomial in theta is real-rooted with every root <= 0 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":{"interlacing":{"from_index":1,"justification":{"premises":["the parent theorem (LiuWangDerivativeLag) proves P_(m-1) weakly interlacing P_m for every m >= 1; T = f(n, theta) = 1 + (2) theta + (1) theta^2 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 >= 1"],"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":"f(n, theta) = 1 + (2) theta + (1) theta^2","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","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 = f(n, theta) = 1 + (2) theta + (1) theta^2, 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 multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(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":0,"interlacing":"weak","interlacing_from":1,"root_region":{"upper":{"closed":true,"value":"0"}},"simple_roots":true,"statement":"every P_n, n >= 0, 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 >= 1 with Q_n(0) != 0, and for n >= 2 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 = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_2 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 1, ..., 1"},"hypotheses":[{"evidence":"checked exactly","name":"seed_roots_negative","statement":"P_1 / x^1 and P_2 / 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_2 / x^1 = deg P_1 / x^1 + 1, with leading coefficients of one sign","status":{"kind":"verified"}},{"evidence":"Bezout test, exact","name":"seed_strict_interlacing","statement":"P_1 / x^1 strictly interlaces P_2 / x^1 (simple real roots, no common root)","status":{"kind":"verified"}},{"evidence":"checked exactly for P_0, ..., P_0","name":"earlier_seeds","statement":"the seed rows P_0, ..., P_0 are each zero or real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"shared degree-law certificate, route recurrence (RealRooted RealRooted.LinRec.natDegree_eq_and_leadingCoeff_pos_of_ratio, RealRooted.ScalarRec.pos_of_ratio): the degree law deg P_n = 0 + 1 (n - 0) with positive leading coefficient for n >= 0 (route recurrence, induction from n = 3); every row is nonzero; lead P_n = 1 for every n >= 1; degree steps {1} from n = 2; the certificate is recomputed when the report is replayed","name":"degree_law","statement":"deg P_n = deg P_(n-1) + 1 with leading coefficients of one sign for every n >= 2","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 >= 3","status":{"kind":"verified"}},{"evidence":"A_n(0) = 1 > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of A_n(0) = 1 at n = 3 + 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":"3","last":null}},"decision":"holds"},"statement":"A_n(0) > 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"V_n = 0","name":"derivative_coefficient_nonpositive","statement":"V_n(x) <= 0 for every x < 0 and every n >= 3","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = -x^2, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 3 + 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":"3","last":null}},"decision":"holds"},"statement":"W_n(x) <= 0 for every x < 0 and every n >= 3","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = -x^2, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 3 + 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":"3","last":null}},"decision":"holds"},"statement":"for every n >= 3 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 >= 1 (x^1 divides P_1 and P_2, 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 >= 3, A_n(x) = (1) + (2) x, V_n(x) = 0, W_n(x) = (-1) x^2 (A_n = a_n + 1 V_n / x for the rule's a_n(x) = (1) + (2) x)","recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","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":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"detail":"no theorem matched (weighted_renewal_window: support must satisfy offset = 1 + ell * t_degree; fixed_delay_predecessor_window: only P_(n-1) and one term w(n) t P_(n-L) are supported)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"HypothesisFailed: P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 2, with a_n = 1, b_n = 2, c_n = 1; Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); open hypotheses: seeds_interlace: deg Q_1 != deg Q_0 + 1 = 1; first_coefficient_positive: not checked: the seeds do not satisfy the seed hypothesis; root_upper_bound: not attempted: the real-rootedness hypotheses do not all hold","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 >= 1 (x^1 divides P_1 and P_2, 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 >= 3, A_n(x) = (1) + (2) x, V_n(x) = 0, W_n(x) = (-1) x^2 (A_n = a_n + 1 V_n / x for the rule's a_n(x) = (1) + (2) x): every P_n, n >= 0, 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 >= 1 with Q_n(0) != 0, and for n >= 2 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 = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_2 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 1, ..., 1","route":"liu_wang_derivative_lag","success":true,"variant":"liu_wang_derivative_lag"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = x^1 Q_n for n >= 1 (x^1 divides P_1 and P_2, 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 >= 3, A_n(x) = (1) + (2) x, V_n(x) = 0, W_n(x) = (-1) x^2 (A_n = a_n + 1 V_n / x for the rule's a_n(x) = (1) + (2) x)","every P_n, n >= 0, 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 >= 1 with Q_n(0) != 0, and for n >= 2 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 = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_2 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 1, ..., 1"],"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 >= 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 = f(n, theta) = 1 + (2) theta + (1) theta^2 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"}