{"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); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Liu--Wang) for n >= 3; the rule is applied from n = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_1, ..., P_2 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 2, ..., 2"},"hypotheses":[{"evidence":"P_2 has degree 0; checked exactly","name":"seed_roots_negative","statement":"P_2 is nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"checked exactly for P_1, ..., P_1","name":"earlier_seeds","statement":"the seed rows P_1, ..., P_1 are nonzero and real-rooted with every root in (-inf, 0)","status":{"kind":"verified"}},{"evidence":"lambda_n = n - 1 > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of lambda_n = n - 1 at n = 3 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["2","1"]},"numerator":["-1","1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] a_n + (deg P_(n-1)) [x^2] b_n > 0 for every n >= 3 (deg P_n = deg P_(n-1) + 1 with leading coefficients of one sign)","status":{"kind":"verified"}},{"evidence":"[x^0] c_1(n, x) is the zero polynomial","name":"derivative_coefficient_vanishes_at_zero","statement":"b_n(0) = 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"b_n(x) = gamma(n) w(x) / f(n) with w(x) = -x^2 + x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 3 + u have one sign)","name":"derivative_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"b_n(x) <= 0 for every x < 0 and every n >= 3","status":{"kind":"verified"}},{"evidence":"a_n(0) = 1 > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of a_n(0) = 1 at n = 3 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"a_n(0) > 0 for every n >= 3","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 3, with a_n(x) = (1) + (2 n - 4) x and b_n(x) = (0) + (1) x + (-1) x^2","recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-4"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","schema":"polytool.recurrence-real-rootedness-theorem.v1","start_delay":1,"status":"proved","theorem":"liu_wang"}}