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so the candidate rows satisfy the input rule at every base n >= 0 (the exceptional bases checked exactly), and agreement on the 1 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":"the schedule d_n depends on n","status":"none"},"lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"lean_route":"reversed","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":"the schedule d_n depends on n","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"x^(d_n) P(1/x), d_n = 0 + 1 (n - 0)","kind":"preserver_image","operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","2"],["-3","-6"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0"],["2"],["-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = x^(d_n) P(1/x), d_n = 0 + 1 (n - 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 x + b_n) P_(n-1) - c_n P_(n-2) for n >= 2, with a_n = (4 n^2 - 6 n + 2)/(n^2 - n), b_n = (2 n^2 - 3 n + 1)/(n^2 - n), c_n = (n^2 - 2 n)/(n^2 - n); open hypotheses: first_coefficient_positive: beta_2 = 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 >= 2, with A_n(x) = ((2 n^2 - 3 n + 1)/(n^2 - n)) + ((4 n^2 - 6 n + 2)/(n^2 - n)) x, V_n(x) = 0, W_n(x) = ((-n^2 + 2 n)/(n^2 - n)); open hypotheses: seed_roots_negative: P_1 is zero, not real-rooted, or has a root >= 0 (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = -n^2 + 2 n is not the zero polynomial; derivative_or_lag_coefficient_negative: W_n(x) = gamma(n) w(x) / f(n) with w(x) = 2, which is positive on (-inf, 0) (no root there, exact): fails at n = 2: gamma(n)/f(n) = (-1/2 n^2 + n)/(n^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: P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 3, with A_n(x) = ((2 n^2 - 3 n + 1)/(n^2 - n)) + ((4 n^2 - 6 n + 2)/(n^2 - n)) x, B_n(x) = ((-n^2 + 2 n)/(n^2 - n)); P_n = x^1 R_n for n >= 1 (x^1 divides P_1 and P_2; R_n satisfies the same rule): every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0] (so its coefficients have one sign); for n >= 2, 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 >= 3; P_n = x^1 R_n for n >= 1 with R_n(0) != 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_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 >= 3, with A_n(x) = ((2 n^2 - 3 n + 1)/(n^2 - n)) + ((4 n^2 - 6 n + 2)/(n^2 - n)) x, B_n(x) = ((-n^2 + 2 n)/(n^2 - n)); P_n = x^1 R_n for n >= 1 (x^1 divides P_1 and P_2; R_n satisfies the same rule)","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0] (so its coefficients have one sign); for n >= 2, 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 >= 3; P_n = x^1 R_n for n >= 1 with R_n(0) != 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_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"},"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"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = x^(d_n) P(1/x), d_n = 0 + 1 (n - 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":"preserver_image"}