{"certification":{"candidate":{"basis":{"family":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"name":"A008277"},"coefficient_family":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","1"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","-4"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},"coefficient_roots":{"parent_proof":{"route":"basis_map_coefficient_roots(liu_wang_three_term)","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 affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"f_polynomial"},{"detail":"no affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"reversal"},{"detail":"no affine degree schedule fits the first parent rows","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 affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 1 row(s)"},{"detail":"no affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"reversal, induction start moved 1 row(s)"},{"detail":"no affine degree schedule fits the first parent rows","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 >= 3, with a_n = (0)/(n^2 - 3 n + 2), b_n = (2 n^2 - 4 n)/(n^2 - 3 n + 2), c_n = (n^2 - n)/(n^2 - 3 n + 2); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_1 + n - 1, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); open hypotheses: leading_coefficient_nonzero: fails at n = 3: f(n) a_n = 0 = 0 there (exact); seeds_interlace: P_1 is zero; first_coefficient_positive: not checked: the seeds do not satisfy the seed hypothesis; recurrence_coefficient_positive: fails at n = 4: the denominator of beta_n = n^4 - 6 n^3 + 11 n^2 - 6 n vanishes there; earlier_seeds: P_0 is zero or not real-rooted with every root in (-inf, 0) (exact); 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":"HypothesisFailed: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 3, with A_n(x) = ((2 n^2 - 4 n)/(n^2 - 3 n + 2)) x, V_n(x) = 0, W_n(x) = ((-n^2 + n)/(n^2 - 3 n + 2)) x^2; open hypotheses: seed_roots_negative: P_1 is zero, not real-rooted, or has a root >= 0 (exact); seed_degree_step: a seed row of the induction is zero; seed_strict_interlacing: the Bezout test fails or the degrees do not differ by one (exact); lag_degree_step_nonnegative: fails at n = 3: mu_n = (-n^2 + n)/(n^2 - 3 n + 2) = -3 there (exact); degree_step_positive: fails at n = 3: lambda_n + mu_n = (n^2 - 3 n)/(n^2 - 3 n + 2) = 0 there (exact); constant_term_positive: fails at n = 3: A_n(0) = (0)/(n^2 - 3 n + 2) = 0 there (exact)","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + gamma(n) (x - L)(x - U) P'_(n-1)","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"induction start moved 1 row(s) past the seeds: Q_n = A_n(x) Q_(n-1) + B_n(x) Q_(n-2) for n >= 4, with A_n(x) = ((2 n^2 - 4 n)/(n^2 - 3 n + 2)), B_n(x) = ((-n^2 + n)/(n^2 - 3 n + 2)); Q_n = P_n / x^(n - 2) for n >= 2, which satisfies the rule below: 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, Q_(n-1) interlaces Q_n (weakly, in the sense of Liu and Wang; no interlacing of the P_n is claimed); 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_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"route":"liu_wang_three_term","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":["multiplier_slope_positive","kappa_nonnegative","initial_row_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":"source (-inf, a + b h + kappa m] = (-inf, 1] with m = 0, target (-inf, 0]"},"identity":{"certificate":{"basis":{"family":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"name":"A008277"},"coefficient_family":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","1"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","-4"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"direction":"input_by_transported","exceptional_bases":[],"input":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["1","3","1"],["-1","-3","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","3","2"],["0","-3","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","1"],["0","0","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["-4","-4"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","-4"],["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"multiplier":["0","1","1"],"quotients":[{"row":[[{"coeff":{"coeffs":[["0"],["1"],["1"]]},"deriv_order":0}]],"shift":0}]},"status":"certified"},"input":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["1","3","1"],["-1","-3","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","3","2"],["0","-3","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","1"],["0","0","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["-4","-4"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","-4"],["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"lift_summary":{"hypotheses":[{"evidence":"exact","name":"multiplier_slope_positive","statement":"b > 0","status":{"kind":"verified"}},{"evidence":"exact","name":"kappa_nonnegative","statement":"kappa >= 0","status":{"kind":"verified"}},{"evidence":"exact root isolation","name":"initial_row_roots","statement":"g = P_0 is nonzero and real-rooted with every root <= 0","status":{"kind":"verified"}}],"identity":{"method":"basis_map_division_input_by_transported","status":"certified"},"input_first_index":0,"input_interlacing":{"reason":"the map theorem does not prove that T_P preserves real-rootedness on the whole line","status":"not_proved"},"lean_replay":"not_attempted","operation":{"basis":"A008277","basis_family":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"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":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","1"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","-4"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = T_P(R_n), T_P(u^k) = P_(1 + k), P the basis family A008277"},"parent_proof":{"route":"basis_map_coefficient_roots(liu_wang_three_term)","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":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)"},"verdict":"proved"},"map_theorem":{"basis":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"basis_name":"A008277","conclusions":[{"hypotheses":["multiplier_slope_positive","kappa_nonnegative","initial_row_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":"source (-inf, a + b h + kappa m] = (-inf, 1] with m = 0, target (-inf, 0]"}],"hypotheses":[{"evidence":"exact","name":"multiplier_slope_positive","statement":"b > 0","status":{"kind":"verified"}},{"evidence":"exact","name":"kappa_nonnegative","statement":"kappa >= 0","status":{"kind":"verified"}},{"evidence":"exact root isolation","name":"initial_row_roots","statement":"g = P_0 is nonzero and real-rooted with every root <= 0","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.rayBasisMap_maps_realRootedIn_iff"],"lean_replay":"not_attempted","schema":"polytool.recurrence-basis-map-theorem.v1","shape":{"a":"1","b":"1","initial_row":["1"],"kappa":"1","kind":"ray_differential","root":"0"},"status":"proved","theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)"},"schema":"polytool.recurrence-basis-map-lift.v1"},"kind":"basis_map"}