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Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","requested_root_region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"3"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"},"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"lift_summary":{"hypotheses":[{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":0,"input_interlacing":{"interlacing":{"from_index":1,"justification":{"premises":["the parent theorem (ThreeTermRecurrence) proves P_(m-1) strictly interlacing P_m for every m >= 1; T = P((1) x + (1)) is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 1"],"theorem":"Hermite--Kakeya--Obreschkoff theorem (N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, 1963; see S. Fisk, Polynomials, Roots, and Interlacing, arXiv:math/0612833): real polynomials f, g have weakly interlacing zeros iff alpha f + beta g is real-rooted or zero for all real alpha, beta; a linear T independent of n that maps every real-rooted polynomial to a real-rooted polynomial or zero therefore maps interlacing pairs to pairs with weakly interlacing zeros (orientation not determined)"},"kind":"weak","orientation":"unordered"},"status":"proved"},"lean_replay":"not_attempted","operation":{"description":"P((1) x + (1))","kind":"preserver_image","operator":{"intercept":"1","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["-2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (1)), for every n >= 0"},"parent_proof":{"route":"three_term_interval","statement":"every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-2, 3)","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, is zero or real-rooted with roots in (-3, 2)","input_root_region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"2"}},"parent_hypothesis":"every parent row used is real-rooted with every root in (-2, 3)","parent_root_region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"3"}},"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically"},"verdict":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_interlacing_report":{"conclusion":{"first_index":0,"interlacing":"strict","interlacing_from":1,"simple_roots":true,"statement":"every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1"},"hypotheses":[{"evidence":"f(n) a_n = 1 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 1 at n = 2 + u have one sign)","name":"leading_coefficient_nonzero","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":"a_n != 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"Q_0 has degree 0, Q_1 degree 1; simple real roots and strict interlacing checked exactly","name":"seeds_interlace","statement":"Q_0 and Q_1 have simple real roots, degrees e and e + 1, and Q_0 strictly interlaces Q_1","status":{"kind":"verified"}},{"evidence":"beta_2 = 1 (exact)","name":"first_coefficient_positive","statement":"beta_2 = c_2 L_0 / (a_2 L_1) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = 1 > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = 1 at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"beta_n = c_n / (a_n a_(n-1)) > 0 for every n >= 3","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n x + b_n) P_(n-1) - c_n P_(n-2) for n >= 2, with a_n = 1, b_n = 0, c_n = 1","recurrence":{"first_index":0,"initial_polynomials":[["1"],["-2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"},"parent_proof":{"attempts":[],"conditional_on":[],"failing_row_index":1,"rows_checked":10,"statement":"parent row P_1 is nonzero and not real-rooted with coefficients of one sign; this parent gives no proof","status":"refuted"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (1)) for every n >= 0, the side conditions hold for every n, and every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-2, 3); so every Q_n, n >= 0, is zero or real-rooted with roots in (-3, 2)","whole_line_parent_proof":{"conclusion":{"first_index":0,"interlacing":"strict","interlacing_from":1,"simple_roots":true,"statement":"every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1"},"hypotheses":[{"evidence":"f(n) a_n = 1 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 1 at n = 2 + u have one sign)","name":"leading_coefficient_nonzero","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":"a_n != 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"Q_0 has degree 0, Q_1 degree 1; simple real roots and strict interlacing checked exactly","name":"seeds_interlace","statement":"Q_0 and Q_1 have simple real roots, degrees e and e + 1, and Q_0 strictly interlaces Q_1","status":{"kind":"verified"}},{"evidence":"beta_2 = 1 (exact)","name":"first_coefficient_positive","statement":"beta_2 = c_2 L_0 / (a_2 L_1) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = 1 > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = 1 at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"beta_n = c_n / (a_n a_(n-1)) > 0 for every n >= 3","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n x + b_n) P_(n-1) - c_n P_(n-2) for n >= 2, with a_n = 1, b_n = 0, c_n = 1","recurrence":{"first_index":0,"initial_polynomials":[["1"],["-2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"}},"kind":"preserver_image"}