{"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) = n != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) = n 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":["3","1"]},"numerator":["0","1"],"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 0 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 = ([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 [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","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: full growth (deg P_n = deg P_(n-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 + 3)/(n) <= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of B_n = (-n + 3)/(n) at n = 3 + u have one sign)","name":"lag_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonpositive","denominator":["0","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3","1"],"shifted_numerator":["0","-1"]},"numerator":["3","-1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"B_n(x) <= 0 for every real x and 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-ratio induction with g(n) = 1 (the exact ratio): g(n) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of g(n) at n = 3 + u have one sign); u_3/u_2 = 1 >= g(3) = 1 exactly; t_n <= 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of t_n at n = 4 + u have one sign); and s_n + t_n / g(n-1) - g(n) >= 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of s_n + t_n / g(n-1) - g(n) at n = 4 + u have one sign)","name":"constant_terms_positive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["0","1"],"method":{"kind":"taylor_shift","shifted_denominator":["4","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"4","last":null}},"decision":"holds"},"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.threeTerm_interlaces_of_eval_nonpos","RealRooted.threeTerm_interlaces","RealRooted.eval_nonpos_of_natDegree_le_two"],"lean_replay":"not_attempted","lean_route":"sign","normal_form":"P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 3, with A_n(x) = ((2 n - 3)/(n)) + ((4 n - 6)/(n)) x, B_n(x) = ((-n + 3)/(n))","recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-3","-6"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["3"],["-1"]]},"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"}}