{"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":[["1"],["3","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","-4","-2"]]},"deriv_order":0,"offset":2,"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 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: coefficient of P_(n-1) must be 1 after normalization; fixed_delay_predecessor_window: coefficient of P_(n-1) must be 1 (+ w(n) t) after normalization)","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 + 1)) P_(n-1) - c_n (x + 1)^2 P_(n-2) for n >= 2, with a_n = 0, b_n = 3, c_n = 2; Q_n(y) = y^(d_n) P_n(1/y - 1), 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: leading_coefficient_nonzero: fails at n = 2: f(n) a_n = 0 = 0 there (exact); seeds_interlace: deg Q_1 != deg Q_0 + 1 = 1; first_coefficient_positive: not checked: the seeds do not satisfy the seed hypothesis; recurrence_coefficient_positive: fails at n = 3: the denominator of beta_n = 2 vanishes there; root_region: the region contains the pole x = -1 of the coordinate change y = 1/(x + 1)","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 >= 2, with A_n(x) = (3) + (3) x, V_n(x) = 0, W_n(x) = (-2) + (-4) x + (-2) x^2; open hypotheses: multipliers_vanish_at_zero: [x^0] c_2(n, x) = -2 is not the zero polynomial; lag_coefficient_nonpositive: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 1 (not a disproof): fails at n = 2: [x^1] W_n = -4 there (exact); derivative_or_lag_coefficient_negative: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 1 (not a disproof): fails at n = 2: [x^1] W_n = -4 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":"P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 2, with A_n(x) = (3) + (3) x, B_n(x) = (-2) + (-4) x + (-2) x^2: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 1, P_(n-1) interlaces P_n (weakly, in the sense of Liu and Wang), and deg P_n - deg P_(n-1) is 0 or 1 for n >= 2","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 2, with A_n(x) = (3) + (3) x, B_n(x) = (-2) + (-4) x + (-2) x^2","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 1, P_(n-1) interlaces P_n (weakly, in the sense of Liu and Wang), and deg P_n - deg P_(n-1) is 0 or 1 for n >= 2"],"theorem":"liu_wang_three_term: L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction; Proposition 3.3 (the case B_n(x) = -(c_n x + d_n)^2)"},"kind":"weak","orientation":"previous_interlaces_next"},"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":[["1"],["3","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","-4","-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"],["6","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-8","-10","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-10","-4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"multiplier":["1"],"quotients":[{"row":[[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}]],"shift":0}]},"status":"certified"},"input":{"first_index":0,"initial_polynomials":[["1"],["6","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-8","-10","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-10","-4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"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":[["1"],["3","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","-4","-2"]]},"deriv_order":0,"offset":2,"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"}