{"kind":"theorem_check","report":{"conclusion":{"first_index":0,"interlacing":"weak","interlacing_from":2,"root_region":{"upper":{"closed":true,"value":"0"}},"simple_roots":false,"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 >= 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; P_n = x^1 R_n for n >= 1 with R_n(0) != 0"},"hypotheses":[{"evidence":"f(n) = n - 2 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) = n - 2 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":["1","1"]},"numerator":["-2","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 / x^1 << P_2 / x^1 checked exactly","name":"seed_rows","statement":"P_1 / x^1 and P_2 / x^1 are nonzero and real-rooted, of degrees e and e or e + 1, and P_1 / x^1 << P_2 / x^1 (weak interlacing)","status":{"kind":"verified"}},{"evidence":"l_n = ([x^1] A_n) l_(n-1) + ([x^2] B_n) l_(n-2) when deg P_(n-1) - deg P_(n-2) = 1 (then deg P_n - deg P_(n-1) = 1): positive-ratio induction with g(n) = (n + 1)/(n) (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 = 4/3 >= g(3) = 4/3 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":"degree_law","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["0","0","4","-4","1"],"method":{"kind":"taylor_shift","shifted_denominator":["64","96","52","12","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"4","last":null}},"decision":"holds"},"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) + ([x^2] B_n) 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 = p_2 x^2 + p_1 x + p_0 with p_2 <= 0, p_0 <= 0 and p_1^2 <= 4 p_2 p_0 (so B_n <= 0 on the real line): p_2 = [x^2] B_n <= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of p_2 = [x^2] B_n at n = 3 + u have one sign); p_0 = [x^0] B_n <= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of p_0 = [x^0] B_n at n = 3 + u have one sign); p_1^2 - 4 p_2 p_0 <= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of p_1^2 - 4 p_2 p_0 at n = 3 + u have one sign)","name":"lag_coefficient_nonpositive","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 / x^1 and P_2 / x^1 have every root < 0","status":{"kind":"verified"}},{"evidence":"positive-ratio induction with g(n) = (n + 1)/(n) (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 = 4/3 >= g(3) = 4/3 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","0","4","-4","1"],"method":{"kind":"taylor_shift","shifted_denominator":["64","96","52","12","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"}},{"evidence":"checked exactly for P_0, ..., P_0","name":"earlier_seeds","statement":"the seed rows P_0, ..., P_0 are each zero or real-rooted with every root in (-inf, 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 - 4)/(n - 2)) + ((2 n - 4)/(n - 2)) x, B_n(x) = ((-n + 2)/(n - 2)) + ((-2 n + 4)/(n - 2)) x + ((-n + 2)/(n - 2)) x^2; P_n = x^1 R_n for n >= 1 (x^1 divides P_1 and P_2; R_n satisfies the same rule)","recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","2"],["0","3","3"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-4"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","4","2"],["-1","-2","-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; Proposition 3.3 (the case B_n(x) = -(c_n x + d_n)^2)","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"liu_wang_three_term"}}