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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":"P_n = (a_n + b_n (x - 1/2)) P_(n-1) - c_n (x - 1/2)^2 P_(n-2) for n >= 2, with a_n = (4 n + 6)/(n + 2), b_n = (4 n + 6)/(n + 2), c_n = (4 n + 4)/(n + 2); Q_n(y) = y^(d_n) P_n(1/y + 1/2), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2): every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)","route":"three_term_recurrence","success":true,"variant":"three_term_recurrence"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = (a_n + b_n (x - 1/2)) P_(n-1) - c_n (x - 1/2)^2 P_(n-2) for n >= 2, with a_n = (4 n + 6)/(n + 2), b_n = (4 n + 6)/(n + 2), c_n = (4 n + 4)/(n + 2); Q_n(y) = y^(d_n) P_n(1/y + 1/2), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2)","every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)"],"theorem":"three_term_recurrence: three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"three_term_recurrence","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","8","4"],["1","18","36","8"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["3","6"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","4","-4"],["-1","4","-4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"direction":"input_by_transported","exceptional_bases":[],"input":{"first_index":0,"initial_polynomials":[["13","20","4"],["63","182","84","8"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-8","-4"],["-1","-8","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-8","-8"],["0","-8","-8"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4"],["0","0","-4"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["9","6"],["6","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6"],["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"multiplier":["4","1"],"quotients":[{"row":[[{"coeff":{"coeffs":[["4"],["1"]]},"deriv_order":0}]],"shift":0}]},"status":"certified"},"input":{"first_index":0,"initial_polynomials":[["13","20","4"],["63","182","84","8"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-8","-4"],["-1","-8","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-8","-8"],["0","-8","-8"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4"],["0","0","-4"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["9","6"],["6","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6"],["0","4"]]},"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, 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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"}