{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1/2"]]},"terms":[{"coeff":{"coeffs":[["0","-1"],["0","3/2"],["0","-1/2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1/2","-1/2"],["0","-1/2","1/2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["0","1"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","0"],["0","-1","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-1"],["1"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1,2],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","2"],["-3","-3"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"],["0","-1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":2,"multiplier":["1/2","-1/2"],"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1/2"]]},"terms":[{"coeff":{"coeffs":[["0","-1"],["0","3/2"],["0","-1/2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1/2","-1/2"],["0","-1/2","1/2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":2,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["-1"],["1"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["0","-1/2"],["0","1/2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","0"],["0","1/2","-1/2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1/2"],["-1/2"]]},"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 = P((1) x + (1)); so the candidate rows satisfy the input rule at every base n >= 2 (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 (LiuWangDerivativeLag) proves P_(m-1) weakly interlacing P_m for every m >= 2; T = P((1) x + (1)) 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":["Polynomial.Splits.comp_of_degree_le_one"],"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":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 = P((1) x + (1)) 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":"P((1) x + (1))","kind":"preserver_image","operator":{"intercept":"1","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1/2"]]},"terms":[{"coeff":{"coeffs":[["0","-1"],["0","3/2"],["0","-1/2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1/2","-1/2"],["0","-1/2","1/2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (1)), 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, -1]","input_root_region":{"upper":{"closed":true,"value":"-1"}},"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":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 >= 3 with Q_n(0) != 0, and for n >= 4 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 = 5 on in the induction (start moved 2 row(s) past the supplied seeds; P_1, ..., P_4 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 2, ..., 3"},"hypotheses":[{"evidence":"f(n) = -1/2 n + 1 != 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of f(n) = -1/2 n + 1 at n = 5 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["-3/2","-1/2"]},"numerator":["1","-1/2"],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 5","status":{"kind":"verified"}},{"evidence":"checked exactly","name":"seed_roots_negative","statement":"P_3 / x^1 and P_4 / 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_4 / x^1 = deg P_3 / x^1 + 1, with leading coefficients of one sign","status":{"kind":"verified"}},{"evidence":"Bezout test, exact","name":"seed_strict_interlacing","statement":"P_3 / x^1 strictly interlaces P_4 / x^1 (simple real roots, no common root)","status":{"kind":"verified"}},{"evidence":"checked exactly for P_1, ..., P_2","name":"earlier_seeds","statement":"the seed rows P_1, ..., P_2 are each zero or real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"lambda_n = (n - 1)/(n - 2) > 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of lambda_n = (n - 1)/(n - 2) at n = 5 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3","1"],"shifted_numerator":["4","1"]},"numerator":["-1","1"],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] A_n + (deg P_(n-1)) [x^2] V_n > 0 for every n >= 5 (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 >= 5","status":{"kind":"verified"}},{"evidence":"A_n(0) = (n - 1)/(n - 2) > 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of A_n(0) = (n - 1)/(n - 2) at n = 5 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3","1"],"shifted_numerator":["4","1"]},"numerator":["-1","1"],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"A_n(0) > 0 for every n >= 5","status":{"kind":"verified"}},{"evidence":"V_n(x) = gamma(n) w(x) / f(n) with w(x) = -1/2 x^2 + 1/2 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (2 n - 2)/(n - 2) >= 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (2 n - 2)/(n - 2) at n = 5 + u have one sign)","name":"derivative_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3","1"],"shifted_numerator":["8","2"]},"numerator":["-2","2"],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"V_n(x) <= 0 for every x < 0 and every n >= 5","status":{"kind":"verified"}},{"evidence":"W_n = 0","name":"lag_coefficient_nonpositive","statement":"W_n(x) <= 0 for every x < 0 and every n >= 5","status":{"kind":"verified"}},{"evidence":"V_n(x) = gamma(n) w(x) / f(n) with w(x) = -1/2 x^2 + 1/2 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (2 n - 2)/(n - 2) > 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (2 n - 2)/(n - 2) at n = 5 + u have one sign)","name":"derivative_or_lag_coefficient_negative","sign_decision":{"certificate":{"condition":"positive","denominator":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3","1"],"shifted_numerator":["8","2"]},"numerator":["-2","2"],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"for every n >= 5 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 = x^1 Q_n for n >= 3 (x^1 divides P_3 and P_4, 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 >= 5, A_n(x) = ((n - 1)/(n - 2)) + ((n^2 - 4 n + 3)/(n - 2)) x, V_n(x) = ((n - 1)/(n - 2)) x + ((-n + 1)/(n - 2)) x^2, W_n(x) = 0 (A_n = a_n + 1 V_n / x for the rule's a_n(x) = ((n^2 - 3 n + 2)/(n - 2)) x)","recurrence":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1/2"]]},"terms":[{"coeff":{"coeffs":[["0","-1"],["0","3/2"],["0","-1/2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1/2","-1/2"],["0","-1/2","1/2"]]},"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":2,"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":"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) = ((0)/(n - 2)) + ((n^2 - 3 n + 2)/(n - 2)) x and b_n(x) = ((0)/(n - 2)) + ((n - 1)/(n - 2)) x + ((-n + 1)/(n - 2)) x^2; open hypotheses: constant_term_positive: fails at n = 3: a_n(0) = (0)/(n - 2) = 0 there (exact)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"induction start moved 2 row(s) past the seeds: P_n = x^1 Q_n for n >= 3 (x^1 divides P_3 and P_4, 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 >= 5, A_n(x) = ((n - 1)/(n - 2)) + ((n^2 - 4 n + 3)/(n - 2)) x, V_n(x) = ((n - 1)/(n - 2)) x + ((-n + 1)/(n - 2)) x^2, W_n(x) = 0 (A_n = a_n + 1 V_n / x for the rule's a_n(x) = ((n^2 - 3 n + 2)/(n - 2)) 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 >= 3 with Q_n(0) != 0, and for n >= 4 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 = 5 on in the induction (start moved 2 row(s) past the supplied seeds; P_1, ..., P_4 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 2, ..., 3","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 >= 3 (x^1 divides P_3 and P_4, 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 >= 5, A_n(x) = ((n - 1)/(n - 2)) + ((n^2 - 4 n + 3)/(n - 2)) x, V_n(x) = ((n - 1)/(n - 2)) x + ((-n + 1)/(n - 2)) x^2, W_n(x) = 0 (A_n = a_n + 1 V_n / x for the rule's a_n(x) = ((n^2 - 3 n + 2)/(n - 2)) 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 >= 3 with Q_n(0) != 0, and for n >= 4 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 = 5 on in the induction (start moved 2 row(s) past the supplied seeds; P_1, ..., P_4 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 2, ..., 3"],"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 = P((1) x + (1)) 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, -1]"},"kind":"preserver_image"}