{"kind":"theorem_check","report":{"conclusion":{"first_index":0,"interlacing":"strict","interlacing_from":4,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":true,"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 >= 4, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 5; the rule is applied from n = 5 on in the induction (start moved 2 row(s) past the supplied seeds; P_0, ..., P_4 enter as seed rows, checked exactly)"},"hypotheses":[{"evidence":"f(n) = n - 2 != 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of f(n) = n - 2 at n = 5 + 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":["-2","1"],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 5","status":{"kind":"verified"}},{"evidence":"checked exactly","name":"seed_roots_negative","statement":"P_3 and P_4 are nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"degrees 3 and 4, leading coefficients of one sign","name":"seed_degree_step","statement":"deg P_4 = deg P_3 + 1, with leading coefficients of one sign","status":{"kind":"verified"}},{"evidence":"Bezout test, exact","name":"seed_strict_interlacing","statement":"P_3 strictly interlaces P_4 (simple real roots, no common root)","status":{"kind":"verified"}},{"evidence":"checked exactly for P_0, ..., P_2","name":"earlier_seeds","statement":"the seed rows P_0, ..., P_2 are each zero or real-rooted with every root in (-inf, 0)","status":{"kind":"verified"}},{"evidence":"lambda_n = (n - 2)/(n - 2) > 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of lambda_n = (n - 2)/(n - 2) at n = 5 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3","1"],"shifted_numerator":["3","1"]},"numerator":["-2","1"],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] A_n + (deg P_(n-1)) [x^2] V_n > 0 for every n >= 5 (deg P_n = deg P_(n-1) + 1 with leading coefficients of one sign)","status":{"kind":"verified"}},{"evidence":"[x^0] c_1 and [x^0] c_2 are zero polynomials","name":"multipliers_vanish_at_zero","statement":"V_n(0) = 0 and W_n(0) = 0 for every n >= 5","status":{"kind":"verified"}},{"evidence":"A_n(0) = (n - 2)/(n - 2) > 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of A_n(0) = (n - 2)/(n - 2) at n = 5 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3","1"],"shifted_numerator":["3","1"]},"numerator":["-2","1"],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"A_n(0) > 0 for every n >= 5","status":{"kind":"verified"}},{"evidence":"V_n(x) = gamma(n) w(x) / f(n) with w(x) = 2 x^2 - 2 x, which is positive on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (-1/2 n + 1)/(n - 2) <= 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (-1/2 n + 1)/(n - 2) at n = 5 + u have one sign)","name":"derivative_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonpositive","denominator":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3","1"],"shifted_numerator":["-3/2","-1/2"]},"numerator":["1","-1/2"],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"V_n(x) <= 0 for every x < 0 and every n >= 5","status":{"kind":"verified"}},{"evidence":"W_n = 0","name":"lag_coefficient_nonpositive","statement":"W_n(x) <= 0 for every x < 0 and every n >= 5","status":{"kind":"verified"}},{"evidence":"V_n(x) = gamma(n) w(x) / f(n) with w(x) = 2 x^2 - 2 x, which is positive on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (-1/2 n + 1)/(n - 2) < 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (-1/2 n + 1)/(n - 2) at n = 5 + u have one sign)","name":"derivative_or_lag_coefficient_negative","sign_decision":{"certificate":{"condition":"negative","denominator":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3","1"],"shifted_numerator":["-3/2","-1/2"]},"numerator":["1","-1/2"],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"for every n >= 5 and x < 0, V_n(x) < 0 or W_n(x) < 0 (V_n < 0 on (-inf, 0) for every n, or W_n < 0 there for every n)","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.derivLag_strictInterl_of_nonnegCoeffs","RealRooted.derivLag_strictInterl_and_noCommonRoot","RealRooted.derivLag_eval_zero_pos"],"lean_replay":"not_attempted","lean_route":"halfGrowth","normal_form":"P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 5, with A_n(x) = ((n - 2)/(n - 2)) + ((n^2 - 2 n)/(n - 2)) x, V_n(x) = ((n - 2)/(n - 2)) x + ((-n + 2)/(n - 2)) x^2, W_n(x) = 0","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"],["1","2","1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","0"],["1","-2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","2"],["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 the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), 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":2,"status":"proved","theorem":"liu_wang_derivative_lag"}}