{"kind":"theorem_check","report":{"conclusion":{"first_index":3,"interlacing":"weak","interlacing_from":4,"root_region":{"upper":{"closed":true,"value":"0"}},"simple_roots":true,"statement":"P_2 is zero; every other P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0] (so its coefficients have one sign); P_n = x^1 Q_n for n >= 3 with Q_n(0) != 0, and for n >= 4 the Q_n have simple roots, Q_(n-1) strictly interlaces Q_n, and P_(n-1) interlaces P_n (Liu--Wang, common root 0); the rule is applied from n = 5 on in the induction (start moved 2 row(s) past the supplied seeds; P_1, ..., P_4 enter as seed rows, checked exactly)"},"hypotheses":[{"evidence":"checked exactly","name":"seed_roots_negative","statement":"P_3 / x^1 and P_4 / x^1 are nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"degrees 0 and 1, leading coefficients of one sign","name":"seed_degree_step","statement":"deg P_4 / x^1 = deg P_3 / x^1 + 1, with leading coefficients of one sign","status":{"kind":"verified"}},{"evidence":"Bezout test, exact","name":"seed_strict_interlacing","statement":"P_3 / x^1 strictly interlaces P_4 / x^1 (simple real roots, no common root)","status":{"kind":"verified"}},{"evidence":"checked exactly for P_1, ..., P_2 (1 zero)","name":"earlier_seeds","statement":"the seed rows P_1, ..., P_2 are each zero or real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"lambda_n = 1 > 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of lambda_n = 1 at n = 5 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["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) = 1 > 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of A_n(0) = 1 at n = 5 + 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":"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) = -x^2 + x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 >= 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 5 + 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":"5","last":null}},"decision":"holds"},"statement":"V_n(x) <= 0 for every x < 0 and every n >= 5","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = -2 x, which is positive on (-inf, 0) (no root there, exact): gamma(n)/f(n) = -1/2 n + 1 <= 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = -1/2 n + 1 at n = 5 + u have one sign)","name":"lag_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonpositive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["-3/2","-1/2"]},"numerator":["1","-1/2"],"range":{"first":"5","last":null}},"decision":"holds"},"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) = -x^2 + x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 > 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 5 + u have one sign)","name":"derivative_or_lag_coefficient_negative","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"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 = x^1 Q_n for n >= 3 (x^1 divides P_3 and P_4, and V_n(0) = 0), with Q_n = A_n(x) Q_(n-1) + V_n(x) Q'_(n-1) + W_n(x) Q_(n-2) for n >= 5, A_n(x) = (1) + (n - 3) x, V_n(x) = (1) x + (-1) x^2, W_n(x) = (n - 2) x (A_n = a_n + 1 V_n / x for the rule's a_n(x) = (n - 2) x)","recurrence":{"first_index":1,"initial_polynomials":[["1"],["0"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2"],["0","1"]]},"deriv_order":0,"offset":2,"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"}}