{"kind":"theorem_check","report":{"conclusion":{"first_index":0,"root_region":{"upper":{"closed":true,"value":"0"}},"simple_roots":false,"statement":"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, Q_(n-1) interlaces Q_n (weakly, in the sense of Liu and Wang; no interlacing of the P_n is claimed)"},"hypotheses":[{"evidence":"degrees 0 and 0; real-rooted and Q_0 << Q_1 checked exactly","name":"seed_rows","statement":"Q_0 and Q_1 are nonzero and real-rooted, of degrees e and e or e + 1, and Q_0 << Q_1 (weak interlacing)","status":{"kind":"verified"}},{"evidence":"l_n = (0) l_(n-1) + ([x^1] B_n) l_(n-2) when deg P_(n-1) - deg P_(n-2) = 0 (then deg P_n - deg P_(n-1) = 1): positive seeds and nonnegative step coefficients: s_n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of s_n at n = 2 + u have one sign); t_n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of t_n at n = 2 + u have one sign); s_n + t_n > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of s_n + t_n at n = 2 + u have one sign)","name":"degree_law_from_step_0","statement":"deg P_n - deg P_(n-1) is 0 or 1 and P_n, P_(n-2) have positive leading coefficients (after the sign normalization) for every n >= 2: degree steps (deg P_(n-1) - deg P_(n-2) -> deg P_n - deg P_(n-1)) 0 -> 1, 1 -> 0; l_n = (0) l_(n-1) + ([x^1] B_n) l_(n-2) when deg P_(n-1) - deg P_(n-2) = 0 (then deg P_n - deg P_(n-1) = 1)","status":{"kind":"verified"}},{"evidence":"l_n = ([x^0] A_n) l_(n-1) + ([x^1] B_n) l_(n-2) when deg P_(n-1) - deg P_(n-2) = 1 (then deg P_n - deg P_(n-1) = 0): positive seeds and nonnegative step coefficients: s_n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of s_n at n = 2 + u have one sign); t_n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of t_n at n = 2 + u have one sign); s_n + t_n > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of s_n + t_n at n = 2 + u have one sign)","name":"degree_law_from_step_1","statement":"deg P_n - deg P_(n-1) is 0 or 1 and P_n, P_(n-2) have positive leading coefficients (after the sign normalization) for every n >= 2: degree steps (deg P_(n-1) - deg P_(n-2) -> deg P_n - deg P_(n-1)) 0 -> 1, 1 -> 0; l_n = ([x^0] A_n) l_(n-1) + ([x^1] B_n) l_(n-2) when deg P_(n-1) - deg P_(n-2) = 1 (then deg P_n - deg P_(n-1) = 0)","status":{"kind":"verified"}},{"evidence":"on x <= 0 (x < 0 and continuity): B_n(x) = gamma(n) w(x) / f(n) with w(x) = x - 4, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 2 + u have one sign)","name":"lag_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"B_n(x) <= 0 for every real x, or for every x <= 0 (with a root location), for every n >= 2","status":{"kind":"verified"}},{"evidence":"checked exactly","name":"seed_roots_negative","statement":"Q_0 and Q_1 have every root < 0","status":{"kind":"verified"}},{"evidence":"positive-ratio induction with g(n) = 2 (the exact ratio): g(n) > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of g(n) at n = 2 + u have one sign); u_2/u_1 = 2 >= g(2) = 2 exactly; t_n <= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of t_n at n = 3 + u have one sign); and s_n + t_n / g(n-1) - g(n) >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of s_n + t_n / g(n-1) - g(n) at n = 3 + u have one sign)","name":"constant_terms_positive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"P_n(0) > 0 for every n >= 2 (normalized rows), with P_n(0) = A_n(0) P_(n-1)(0) + B_n(0) P_(n-2)(0)","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.MaWang.strictInterl_of_interlaces_evalCoeff_nonpos"],"lean_replay":"not_attempted","normal_form":"Q_n = A_n(x) Q_(n-1) + B_n(x) Q_(n-2) for n >= 2, with A_n(x) = (4), B_n(x) = (-4) + (1) x; Q_n = x^(n - 0) P_n(1/x) for n >= 0 (the reversal at the nominal degree deg P_0 + n - 0, an upper bound for deg P_n), which satisfies the rule below","recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"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","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"liu_wang_three_term"}}