{"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":"f(n) = 2 n^2 - 3 n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = 2 n^2 - 3 n at n = 2 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["2","5","2"]},"numerator":["0","-3","2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 2","status":{"kind":"verified"}},{"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 = ([x^1] A_n) l_(n-1) + (0) 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 -> 1; l_n = ([x^1] A_n) l_(n-1) + (0) 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^1] A_n) l_(n-1) + (0) l_(n-2) when deg P_(n-1) - deg P_(n-2) = 1 (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_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 -> 1; l_n = ([x^1] A_n) l_(n-1) + (0) l_(n-2) when deg P_(n-1) - deg P_(n-2) = 1 (then deg P_n - deg P_(n-1) = 1)","status":{"kind":"verified"}},{"evidence":"B_n = (-n^2 + 5/2 n - 1)/(n^2 - 3/2 n) <= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of B_n = (-n^2 + 5/2 n - 1)/(n^2 - 3/2 n) at n = 2 + u have one sign)","name":"lag_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonpositive","denominator":["0","-3/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["1","5/2","1"],"shifted_numerator":["0","-3/2","-1"]},"numerator":["-1","5/2","-1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"B_n(x) <= 0 for every real x and 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) = (n)/(n - 1) (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":["0","0","9/4","-15/2","37/4","-5","1"],"method":{"kind":"taylor_shift","shifted_denominator":["81","243","1197/4","387/2","277/4","13","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) = ((2 n^2 - 4 n + 2)/(n^2 - 3/2 n)) + ((4 n^2 - 8 n + 3)/(n^2 - 3/2 n)) x, B_n(x) = ((-n^2 + 5/2 n - 1)/(n^2 - 3/2 n)); 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","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","4"],["-16","-8"],["8","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","5"],["0","0","-2"]]},"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"}}