{"kind":"theorem_check","report":{"conclusion":{"first_index":0,"interlacing":"weak","interlacing_from":2,"root_region":{"lower":{"closed":true,"value":"-4"}},"simple_roots":false,"statement":"every P_n, n >= 0, is nonzero and real-rooted with every root in [-4, +inf); 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 - 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 1 and 2; 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":"[t^1] B_n(-4 + t) <= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of [t^1] B_n(-4 + t) at n = 3 + u have one sign)","name":"lag_coefficient_nonpositive","statement":"B_n(x) <= 0 for every x in [-4, +inf) and every n >= 3","status":{"kind":"verified"}},{"evidence":"[t^1] A_n(-4 - t) <= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of [t^1] A_n(-4 - t) at n = 3 + u have one sign); A_n(-5) < 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of A_n(-5) at n = 3 + u have one sign); [t^1] B_n(-4 - t) >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of [t^1] B_n(-4 - t) at n = 3 + u have one sign)","name":"window_lower_end","statement":"deg P_n = deg P_(n-1) + 1, A_n(x) < 0 and B_n(x) >= 0 for every x < -4 and every n >= 3 (no row gets a root < -4)","status":{"kind":"verified"}},{"evidence":"checked exactly","name":"seed_roots_in_window","statement":"P_1 and P_2 have every root in [-4, +inf)","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 [-4, +inf)","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.threeTerm_interlaces_of_window"],"lean_replay":"not_attempted","normal_form":"P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 3, with A_n(x) = ((4 n - 8)/(n - 2)) + ((n - 2)/(n - 2)) x, B_n(x) = ((-4 n + 8)/(n - 2)) + ((-n + 2)/(n - 2)) x","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"],["2","4","1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-8","-2"],["4","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","2"],["-4","-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"}}