{"kind":"theorem_check","report":{"conclusion":{"first_index":1,"interlacing":"weak","interlacing_from":2,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 1, 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"},"hypotheses":[{"evidence":"f(n) = 1/3 n + 1 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) = 1/3 n + 1 at n = 3 + 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","1/3"]},"numerator":["1","1/3"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 3","status":{"kind":"verified"}},{"evidence":"degrees 1 and 1; real-rooted and P_1 << P_2 checked exactly","name":"seed_rows","statement":"P_1 and P_2 are nonzero and real-rooted, of degrees e and e or e + 1, and P_1 << P_2 (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 [3, +inf) (exact; Taylor shift: the coefficients of s_n at n = 3 + u have one sign); 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); s_n + t_n > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of s_n + t_n at n = 3 + 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 >= 3: 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 [3, +inf) (exact; Taylor shift: the coefficients of s_n at n = 3 + u have one sign); 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); s_n + t_n > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of s_n + t_n at n = 3 + 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 >= 3: 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) = 1/3 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (3 n^3 + 6 n^2 + 3 n)/(n + 3) >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (3 n^3 + 6 n^2 + 3 n)/(n + 3) at n = 3 + u have one sign)","name":"lag_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["3","1"],"method":{"kind":"taylor_shift","shifted_denominator":["6","1"],"shifted_numerator":["144","120","33","3"]},"numerator":["0","3","6","3"],"range":{"first":"3","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 >= 3","status":{"kind":"verified"}},{"evidence":"checked exactly","name":"seed_roots_negative","statement":"P_1 and P_2 have every root < 0","status":{"kind":"verified"}},{"evidence":"positive seeds and nonnegative step coefficients: s_n >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of s_n at n = 3 + u have one sign); 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); s_n + t_n > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of s_n + t_n at n = 3 + u have one sign)","name":"constant_terms_positive","statement":"P_n(0) > 0 for every n >= 3 (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":"P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 3, with A_n(x) = ((4 n + 6)/(n + 3)), B_n(x) = ((n^3 + 2 n^2 + n)/(n + 3)) x","recurrence":{"first_index":1,"initial_polynomials":[["5","1"],["14","10"]],"recurrence":{"denominator":{"coeffs":[["1"],["1/3"]]},"terms":[{"coeff":{"coeffs":[["2"],["4/3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","1/3"],["0","2/3"],["0","1/3"]]},"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"}}