{"kind":"theorem_check","report":{"conclusion":{"first_index":0,"interlacing":"weak","interlacing_from":1,"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]; deg P_n = deg P_(n-1) + 1 for n >= 2; P_(n-1) interlaces P_n (weakly: P_n = a_n P_(n-1) with a_n linear adds one real root) for n >= 1"},"hypotheses":[{"evidence":"f(n) = n - 1 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = n - 1 at n = 2 + 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":["-1","1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 2","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_1 has degree 1; checked exactly","name":"start_row_in_region","statement":"P_1 is nonzero and real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"checked exactly for P_0, ..., P_0","name":"earlier_seeds_in_region","statement":"every nonzero seed row among P_0, ..., P_0 is real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"delta = 1, deg P_(n-1) = n - 1: Lambda(n, deg P_(n-1)) = 2 n - 2 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of Lambda(n, deg P_(n-1)) = 2 n - 2 at n = 2 + u have one sign)","name":"degree_step","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["2","2"]},"numerator":["-2","2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"deg P_n = deg P_(n-1) + delta for every n >= 2: 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 + 1 is real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}}],"iterated_preserver":{"degree_step":1,"induction_start":1,"multiplier_fixed_factor":["1","1"],"multiplier_index_factor":[{"denominator":["-1","1"],"numerator":["-2","2"]}],"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 + 1) (((-2 + 2 n)/(-1 + n)))","region":{"upper":{"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 >= 2, with a_n(x) = (x + 1) (((-2 + 2 n)/(-1 + n))) and S_n = the identity","recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","2"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-2"],["2","2"]]},"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","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"iterated_preserver"}}