{"kind":"theorem_check","report":{"conclusion":{"first_index":1,"interlacing":"weak","interlacing_from":2,"root_region":{"lower":{"closed":true,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in [0, inf); deg P_n = deg P_(n-1) + 1 for n >= 3; P_(n-1) interlaces P_n (weakly: P_n = a_n P_(n-1) with a_n linear adds one real root) for n >= 2"},"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":"q(n) delta_S(n) sum_d c_d(n, x) D^d = f(n) (g H)(n, x) o S_num holds exactly in the Ore algebra (S_n = the identity)","name":"operator_identity","statement":"the step operator T_n of the rule equals a_n(x) S_n","status":{"kind":"verified"}},{"evidence":"P_2 has degree 1; checked exactly","name":"start_row_in_region","statement":"P_2 is nonzero and real-rooted with every root in [0, inf)","status":{"kind":"verified"}},{"evidence":"checked exactly for P_1, ..., P_1","name":"earlier_seeds_in_region","statement":"every nonzero seed row among P_1, ..., P_1 is real-rooted with every root in [0, inf)","status":{"kind":"verified"}},{"evidence":"delta = 1, deg P_(n-1) = n - 2: Lambda(n, deg P_(n-1)) = -n + 2 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of Lambda(n, deg P_(n-1)) = -n + 2 at n = 3 + u have one sign)","name":"degree_step","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":"deg P_n = deg P_(n-1) + delta for every n >= 3: the coefficient Lambda(n, e) of x^(e + delta) in f(n) T_n(x^e) is nonzero at e = deg P_(n-1)","status":{"kind":"verified"}},{"evidence":"exact check of the fixed polynomial","name":"multiplier_fixed_factor_in_region","statement":"the multiplier factor g(x) = x - 2 is real-rooted with every root in [0, inf)","status":{"kind":"verified"}}],"iterated_preserver":{"degree_step":1,"induction_start":2,"multiplier_fixed_factor":["-2","1"],"multiplier_index_factor":[{"denominator":["-2","1"],"numerator":["2","-1"]}],"operator_identity":"f(n) T_n = sum_d c_d(n, x) D^d equals f(n) a_n(x) S_n with a_n(x) = (x - 2) (((2 + -1 n)/(-2 + n)))","region":{"lower":{"closed":true,"value":"0"}},"start_degree":1},"lean_replay":"not_attempted","normal_form":"P_n = a_n(x) S_n(P_(n-1)) for n >= 3, with a_n(x) = (x - 2) (((2 + -1 n)/(-2 + n))) and S_n = the identity","recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","-1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","2"],["2","-1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; here S_n is the identity; for R = [0, inf) the reflection x -> -x maps the rows to rows with every root in (-inf, 0]","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"iterated_preserver"}}