{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["0"],["4"],["18","18"]],"recurrence":{"denominator":{"coeffs":[["0"],["-2"],["5"],["-4"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["0","0","-1"],["0","0","-2"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4"],["0","2"],["0","-6"],["0","-2"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"existing definition"},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0","-4"],["0","-4","-4"],["0","-4","-1"],["0","-1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0"],["-1","1"],["0","1"],["1","0"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["0","0"],["2","-1"],["-3","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["0","-1","1"],["0","-1","2"],["0","1","1"],["0","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","1","1"],"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["0"],["4"],["18","18"]],"recurrence":{"denominator":{"coeffs":[["0"],["-2"],["5"],["-4"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["0","0","-1"],["0","0","-2"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4"],["0","2"],["0","-6"],["0","-2"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":1,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-18","1"],["9","-8"],["0","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-9","5"],["0","0","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0"],["1"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0"],["0","1","-2","1"],["0","-1","0","1"],["0","-1","2","0"],["0","1","0","0"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","1","-2","1"],["0","0","0","-2","2"],["0","0","-2","2","1"],["0","0","0","2","0"],["0","0","1","0","0"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","-18","54","-54","18"],["0","0","-9","81","-135","63"],["0","0","54","-81","-54","81"],["0","0","27","-162","108","45"],["0","0","-54","0","108","9"],["0","0","-27","81","27","0"],["0","0","18","27","0","0"],["0","0","9","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","-18","48","-42","12"],["0","0","0","-27","110","-127","44"],["0","0","0","27","21","-115","61"],["0","0","0","54","-111","-1","39"],["0","0","0","0","-78","47","11"],["0","0","0","-27","0","20","1"],["0","0","0","-9","9","2","0"],["0","0","0","0","1","0","0"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","162","-342","222","-42"],["0","0","243","-711","553","-109"],["0","0","-243","54","197","-36"],["0","0","-486","918","-498","138"],["0","0","0","432","-465","171"],["0","0","243","-189","-69","78"],["0","0","81","-144","45","15"],["0","0","0","-18","14","1"],["0","0","0","0","1","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","162","-360","258","-60"],["0","0","0","405","-1098","901","-232"],["0","0","0","162","-1017","1134","-349"],["0","0","0","-324","-45","564","-256"],["0","0","0","-324","387","27","-94"],["0","0","0","-81","171","-60","-16"],["0","0","0","0","18","-15","-1"],["0","0","0","0","0","-1","0"]]},"deriv_order":1}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["-2","1"],["3","-1"],["-1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-1","1"],["0","1","0"]]},"deriv_order":1,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0"],["0","-9","14","-5"],["0","0","10","-6"],["0","9","-5","-1"],["0","0","-1","0"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","1","-2","1"],["0","0","0","-2","2"],["0","0","-2","2","1"],["0","0","0","2","0"],["0","0","1","0","0"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","2","-6","6","-2"],["0","0","1","-9","15","-7"],["0","0","-6","9","6","-9"],["0","0","-3","18","-12","-5"],["0","0","6","0","-12","-1"],["0","0","3","-9","-3","0"],["0","0","-2","-3","0","0"],["0","0","-1","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","2","-4","2"],["0","0","0","2","-1","-8","7"],["0","0","0","3","-13","3","9"],["0","0","0","-3","-10","17","5"],["0","0","0","-6","8","13","1"],["0","0","0","0","11","3","0"],["0","0","0","3","3","0","0"],["0","0","0","1","0","0","0"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","-8","8"],["0","0","-18","38","-58","46"],["0","0","-27","79","-111","103"],["0","0","27","-6","-42","115"],["0","0","54","-102","84","67"],["0","0","0","-48","96","19"],["0","0","-27","21","35","2"],["0","0","-9","16","4","0"],["0","0","0","2","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","-18","28","-10"],["0","0","0","-18","-17","80","-37"],["0","0","0","-45","72","59","-52"],["0","0","0","-18","132","-25","-34"],["0","0","0","36","63","-49","-10"],["0","0","0","36","-6","-19","-1"],["0","0","0","9","-9","-2","0"],["0","0","0","0","-1","0","0"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","-1","1"],["0","1","0"]]},"start_base":1}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","36"],["0","0","60"],["0","0","37"],["0","0","10"],["0","0","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0"],["0","-18"],["0","-30"],["0","-14"],["0","-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0"],["2"],["5"],["4"],["1"]]},"deriv_order":0,"shift":2}]},"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 1) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = P((1) x + (0)); so the candidate rows satisfy the input rule at every base n >= 1 (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 >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"none"},"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":0,"input_interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"P((1) x + (0))","kind":"preserver_image","operator":{"intercept":"0","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["4"],["18","18"]],"recurrence":{"denominator":{"coeffs":[["0"],["-2"],["5"],["-4"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["0","0","-1"],["0","0","-2"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4"],["0","2"],["0","-6"],["0","-2"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (0)), for every n >= 0"},"parent_proof":{"route":"liu_wang_three_term","statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]","input_root_region":{"upper":{"closed":true,"value":"0"}},"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":["the division remainder is removed by row annihilators of the shifted parent"],"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":"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":"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^4 - 4 n^3 + 5 n^2 - 2 n), b_n = (2 n^4 - 2 n^3 - 6 n^2 + 2 n + 4)/(n^4 - 4 n^3 + 5 n^2 - 2 n), c_n = (n^4 + 2 n^3 + n^2)/(n^4 - 4 n^3 + 5 n^2 - 2 n); 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); first_coefficient_positive: a_3 or c_3 is undefined or a_3 = 0; recurrence_coefficient_positive: fails at n = 4: the denominator of beta_n = n^8 - 6 n^7 + 8 n^6 + 10 n^5 - 21 n^4 - 4 n^3 + 12 n^2 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^4 - 2 n^3 - 6 n^2 + 2 n + 4)/(n^4 - 4 n^3 + 5 n^2 - 2 n)) x, V_n(x) = 0, W_n(x) = ((-n^4 - 2 n^3 - n^2)/(n^4 - 4 n^3 + 5 n^2 - 2 n)) x^2; open hypotheses: lag_degree_step_nonnegative: fails at n = 3: mu_n = (-n^4 - 2 n^3 - n^2)/(n^4 - 4 n^3 + 5 n^2 - 2 n) = -12 there (exact); degree_step_positive: fails at n = 3: lambda_n + mu_n = (n^4 - 4 n^3 - 7 n^2 + 2 n + 4)/(n^4 - 4 n^3 + 5 n^2 - 2 n) = -20/3 there (exact); constant_term_positive: fails at n = 3: A_n(0) = (0)/(n^4 - 4 n^3 + 5 n^2 - 2 n) = 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^4 - 2 n^3 - 6 n^2 + 2 n + 4)/(n^4 - 4 n^3 + 5 n^2 - 2 n)), B_n(x) = ((-n^4 - 2 n^3 - n^2)/(n^4 - 4 n^3 + 5 n^2 - 2 n)); Q_n = x^(n - 1) P_n(1/x) for n >= 2 (the reversal at the nominal degree deg P_2 + n - 2, an upper bound for deg P_n), which satisfies the rule below: P_0 is 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"},"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 + (0)) for every n >= 0, the side conditions hold for every n, and every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]"},"kind":"definition_equality"}