{"kind":"theorem_check","report":{"conclusion":{"first_index":1,"root_region":{"upper":{"closed":true,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 2"},"hypotheses":[{"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 = phi(n, D) = 1 + (2) D + (1) D^2)","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 0; 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":"delta = 1, deg P_(n-1) = n - 2: Lambda(n, deg P_(n-1)) = 1 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of Lambda(n, deg P_(n-1)) = 1 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":["1"]},"numerator":["1"],"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"}},{"evidence":"the polynomial with ascending coefficients [1, 2, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"derivative_polynomial_real_rooted","statement":"the polynomial in D is real-rooted for every n >= n0","status":{"kind":"verified"}},{"evidence":"the menu image of (-inf, 0] lies in (-inf, 0]","name":"preserver_keeps_region","statement":"S_n maps every polynomial that is zero or real-rooted with every root in (-inf, 0] to zero or such a polynomial, for every n >= 2","status":{"kind":"verified"}}],"iterated_preserver":{"degree_step":1,"induction_start":1,"multiplier_fixed_factor":["1","1"],"multiplier_index_factor":[{"denominator":["1"],"numerator":["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 + 1) ((1))","preserver":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"region":{"upper":{"closed":true,"value":"0"}},"start_degree":0},"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"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) ((1)) and S_n = phi(n, D) = 1 + (2) D + (1) D^2","recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":2,"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; S_n = phi(n, D) = 1 + (2) D + (1) D^2: Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"iterated_preserver"}}