{"certification":{"candidate":{"basis":{"family":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"name":"A007318"},"coefficient_family":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","4"],["-16","-8"],["8","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","5"],["0","0","-2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},"coefficient_roots":{"parent_proof":{"route":"basis_map_coefficient_roots(liu_wang_derivative_lag)","statement":"every coefficient row R_n, n >= 0, is zero or real-rooted with every root in (-inf, 0]","status":"proved"},"source_interval":{"upper":{"closed":true,"value":{"lower_bound":"1","radicand":"1","rational_part":"1","sqrt_coefficient":"0","text":"1","upper_bound":"1"}}},"target_interval":{"upper":{"closed":true,"value":"0"}},"upper_bound":{"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":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 3 row(s)"},{"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 = (4 n^2 - 8 n + 3)/(n^2 - 3/2 n), b_n = (2 n^2 - 4 n + 2)/(n^2 - 3/2 n), c_n = (n^2 - 5/2 n + 1)/(n^2 - 3/2 n); 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) = ((4 n^2 - 8 n + 3)/(n^2 - 3/2 n)) + ((2 n^2 - 4 n + 2)/(n^2 - 3/2 n)) x, V_n(x) = 0, W_n(x) = ((-n^2 + 5/2 n - 1)/(n^2 - 3/2 n)) x^2 (A_n = a_n + 1 V_n / x for the rule's a_n(x) = ((4 n^2 - 8 n + 3)/(n^2 - 3/2 n)) + ((2 n^2 - 4 n + 2)/(n^2 - 3/2 n)) 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) = ((4 n^2 - 8 n + 3)/(n^2 - 3/2 n)) + ((2 n^2 - 4 n + 2)/(n^2 - 3/2 n)) x, V_n(x) = 0, W_n(x) = ((-n^2 + 5/2 n - 1)/(n^2 - 3/2 n)) x^2 (A_n = a_n + 1 V_n / x for the rule's a_n(x) = ((4 n^2 - 8 n + 3)/(n^2 - 3/2 n)) + ((2 n^2 - 4 n + 2)/(n^2 - 3/2 n)) 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"}},"conclusion":{"hypotheses":["initial_row_nonzero","initial_row_real_rooted","initial_row_nonpositive_roots"],"input_interval":{"upper":{"closed":true,"value":{"lower_bound":"1","radicand":"1","rational_part":"1","sqrt_coefficient":"0","text":"1","upper_bound":"1"}}},"nonnegative_coefficients":true,"output_interval":{"upper":{"closed":true,"value":{"lower_bound":"0","radicand":"1","rational_part":"0","sqrt_coefficient":"0","text":"0","upper_bound":"0"}}},"sharp":true,"statement":"an input root r <= 1 gives the root (r - a)/d <= 0; a larger root gives a positive root"},"identity":{"certificate":{"basis":{"family":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"name":"A007318"},"coefficient_family":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","4"],["-16","-8"],["8","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","5"],["0","0","-2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"direction":"input_by_transported","exceptional_bases":[],"input":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["-2","-4","-2"],["5","10","5"],["-2","-4","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["10","4"],["-24","-8"],["12","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"multiplier":["2","5","2"],"quotients":[{"row":[[{"coeff":{"coeffs":[["2"],["5"],["2"]]},"deriv_order":0}]],"shift":0}]},"status":"certified"},"input":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["-2","-4","-2"],["5","10","5"],["-2","-4","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["10","4"],["-24","-8"],["12","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"lift_summary":{"hypotheses":[{"evidence":"exact","name":"initial_row_nonzero","statement":"g = P_0 is nonzero","status":{"kind":"verified"}},{"evidence":"exact root isolation","name":"initial_row_real_rooted","statement":"g is real-rooted","status":{"kind":"verified"}},{"evidence":"exact root isolation","name":"initial_row_nonpositive_roots","statement":"every root of g is <= 0","status":{"kind":"verified"}}],"identity":{"method":"basis_map_division_input_by_transported","status":"certified"},"input_first_index":0,"input_interlacing":{"interlacing":{"from_index":1,"justification":{"premises":["Q_n = T_P(R_n), T_P(u^k) = P_(0 + k), P the basis family A007318 (identity certified)","every coefficient row R_n, n >= 0, is zero or real-rooted with every root in (-inf, 0] (proved, route basis_map_coefficient_roots(liu_wang_derivative_lag))","the parent route proves P_(m-1) << P_m (weakly) for every m >= 1: 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)","g = P_0 is nonzero (verified: exact)","g is real-rooted (verified: exact root isolation)","every root of g is <= 0 (verified: exact root isolation)","T_P (T_P(u^k) = P_(0 + k), P = A007318) does not depend on n and maps every real-rooted polynomial to a real-rooted polynomial or zero (map theorem: geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary))","every alpha P_(m-1) + beta P_m is real-rooted or zero (forward direction), so is its image, and the converse direction gives weakly interlacing zeros of the images; a nonzero gauge moves no zero","hence Q_(n-1) and Q_n interlace (orientation not determined) (weakly) 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":{"basis":"A007318","basis_family":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"kind":"basis_map","source_interval":"(-inf, 1]","target_interval":"(-inf, 0]"},"parent":{"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","4"],["-16","-8"],["8","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","5"],["0","0","-2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = T_P(R_n), T_P(u^k) = P_(0 + k), P the basis family A007318"},"parent_proof":{"route":"basis_map_coefficient_roots(liu_wang_derivative_lag)","statement":"every coefficient row R_n, n >= 0, is zero or real-rooted with every root in (-inf, 0]","status":"proved"},"root_transfer":{"input_conclusion":"every input row Q_n, n >= 0, is zero or real-rooted with every root in (-inf, 0] (nonnegative coefficients when the coefficient row has a positive leading coefficient)","input_root_region":{"upper":{"closed":true,"value":"0"}},"parent_hypothesis":"every coefficient row R_n is zero or real-rooted with every root in the source interval of the map theorem","parent_root_region":{"upper":{"closed":true,"value":"0"}},"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)"},"verdict":"proved"},"map_theorem":{"basis":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"basis_name":"A007318","conclusions":[{"hypotheses":["initial_row_nonzero","initial_row_real_rooted"],"input_interval":{},"nonnegative_coefficients":false,"output_interval":{},"sharp":true,"statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},{"hypotheses":["initial_row_nonzero","initial_row_real_rooted","initial_row_nonpositive_roots"],"input_interval":{"upper":{"closed":true,"value":{"lower_bound":"1","radicand":"1","rational_part":"1","sqrt_coefficient":"0","text":"1","upper_bound":"1"}}},"nonnegative_coefficients":true,"output_interval":{"upper":{"closed":true,"value":{"lower_bound":"0","radicand":"1","rational_part":"0","sqrt_coefficient":"0","text":"0","upper_bound":"0"}}},"sharp":true,"statement":"an input root r <= 1 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"hypotheses":[{"evidence":"exact","name":"initial_row_nonzero","statement":"g = P_0 is nonzero","status":{"kind":"verified"}},{"evidence":"exact root isolation","name":"initial_row_real_rooted","statement":"g is real-rooted","status":{"kind":"verified"}},{"evidence":"exact root isolation","name":"initial_row_nonpositive_roots","statement":"every root of g is <= 0","status":{"kind":"verified"}}],"lean_replay":"not_attempted","schema":"polytool.recurrence-basis-map-theorem.v1","shape":{"constant":"1","initial_row":["1"],"kind":"geometric","slope":"1"},"status":"proved","theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)"},"schema":"polytool.recurrence-basis-map-lift.v1"},"kind":"basis_map"}