{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["6","60","144","0"],["5","74","196","4"],["1","27","84","4"],["0","3","12","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","-40","-312","-336","-64"],["0","3","24","-92","-196","-32"],["0","1","21","36","-20","0"],["0","0","3","12","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","-3","-50","-228","-168","-32"],["0","0","-1","-25","-144","-100","-16"],["0","0","0","-3","-24","-12","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-6","-24","0","0"],["-5","-26","44","8"],["-1","-15","30","8"],["0","-3","6","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-6","-24","0","0"],["0","-5","-26","44","8"],["0","-1","-15","30","8"],["0","0","-3","6","0"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2","-10","-20","0"],["-3","-5","2","-8"],["-1","-6","12","8"],["0","-3","6","0"]]},"terms":[{"coeff":{"coeffs":[["-2","-10","-20","0"],["-3","-29","-64","4"],["-1","-18","-48","-4"],["0","-3","-12","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","28","108","80","16"],["0","1","19","96","76","16"],["0","0","3","24","12","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-4","-12"],["3","-18"],["3","-6"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","8","16","-8"],["-1","-6","12","8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","1","6","-12","-8"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["4"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","-96","-1920","-13056","-33792","-26112","-6144","0","0","0","0"],["0","0","-176","-3424","-20928","-31360","66304","93696","31744","2048","0","0"],["0","0","-120","-2416","-14272","-8064","107392","24832","-79872","-40960","-4096","0"],["0","0","-36","-856","-5504","704","64000","-37120","-72704","-1024","13312","2048"],["0","0","-4","-156","-1224","816","18912","-24768","-20352","17664","13824","2048"],["0","0","0","-12","-120","144","2208","-4416","-1152","3840","1536","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-96","-1920","-13056","-33792","-26112","-6144","0","0","0","0"],["0","0","0","-176","-3424","-20928","-31360","66304","93696","31744","2048","0","0"],["0","0","0","-120","-2416","-14272","-8064","107392","24832","-79872","-40960","-4096","0"],["0","0","0","-36","-856","-5504","704","64000","-37120","-72704","-1024","13312","2048"],["0","0","0","-4","-156","-1224","816","18912","-24768","-20352","17664","13824","2048"],["0","0","0","0","-12","-120","144","2208","-4416","-1152","3840","1536","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","-512","-12288","-104448","-360448","-417792","-196608","-32768","0","0","0"],["0","0","0","-768","-16896","-119808","-227328","454656","958464","540672","98304","0","0"],["0","0","0","-384","-7680","-43008","6144","454656","24576","-688128","-491520","-98304","0"],["0","0","0","-64","-1152","-4608","15360","73728","-147456","-122880","147456","147456","32768"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","-768","-14848","-70656","190464","1896448","2629632","1327104","229376","0","0","0"],["0","0","0","-2304","-43776","-204288","470016","4036608","946176","-4497408","-3391488","-688128","0","0"],["0","0","0","-2688","-49536","-213504","583680","3594240","-3919872","-6905856","294912","2555904","688128","0"],["0","0","0","-1536","-27200","-102528","400896","1661952","-4055040","-1916928","4399104","2506752","-245760","-229376"],["0","0","0","-432","-7296","-22464","138240","382464","-1548288","276480","1916928","-110592","-884736","-245760"],["0","0","0","-48","-768","-1728","18432","32256","-221184","129024","294912","-110592","-196608","-49152"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","256","5888","42496","41984","-702464","-2633728","-3006464","-1392640","-229376","0","0","0","0"],["0","0","0","512","11904","89088","144384","-872448","-2457600","1720320","5554176","3342336","622592","0","0","0"],["0","0","0","384","9024","67968","105984","-691200","-1585152","2875392","3022848","-1622016","-2113536","-491520","0","0"],["0","0","0","128","3040","22400","21120","-328704","-537600","2015232","970752","-2113536","-1613824","-229376","32768","0"],["0","0","0","16","384","2688","-512","-57600","-36864","503808","-147456","-921600","-32768","688128","393216","65536"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","-128","-3840","-43008","-208896","-307200","614400","1671168","1376256","491520","65536","0","0","0"],["0","0","0","0","-192","-5376","-52992","-184320","165888","1843200","663552","-2949120","-3391488","-1376256","-196608","0","0"],["0","0","0","0","-96","-2496","-21120","-39168","267264","755712","-1511424","-2138112","1253376","2703360","1277952","196608","0"],["0","0","0","0","-16","-384","-2688","512","57600","36864","-503808","147456","921600","32768","-688128","-393216","-65536"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","-384","-7424","-37888","34816","450560","-278528","-704512","-32768","229376","65536","0"],["0","0","0","-960","-17536","-74496","196608","1081344","-1904640","-1523712","1769472","1425408","98304","-65536"],["0","0","0","-864","-14976","-51840","248832","857088","-2654208","-331776","3096576","663552","-884736","-294912"],["0","0","0","-336","-5568","-15552","115200","271872","-1327104","460800","1695744","-331776","-933888","-245760"],["0","0","0","-48","-768","-1728","18432","32256","-221184","129024","294912","-110592","-196608","-49152"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","128","2816","19200","21504","-184320","-307200","516096","638976","-98304","-262144","-65536","0","0"],["0","0","0","192","4288","29440","26880","-344064","-497664","1523712","811008","-1228800","-737280","65536","65536","0"],["0","0","0","96","2208","15360","8064","-239616","-285696","1511424","258048","-1695744","-712704","196608","98304","0"],["0","0","0","16","384","2688","-512","-57600","-36864","503808","-147456","-921600","-32768","688128","393216","65536"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","-64","-1792","-17664","-61440","55296","614400","221184","-983040","-1130496","-458752","-65536","0","0"],["0","0","0","0","-64","-1664","-14080","-26112","178176","503808","-1007616","-1425408","835584","1802240","851968","131072","0"],["0","0","0","0","-16","-384","-2688","512","57600","36864","-503808","147456","921600","32768","-688128","-393216","-65536"]]},"deriv_order":2}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0"],["-6","12"],["-9","18"],["-3","6"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["2","-8","-16","0"],["3","8","-8","8"],["1","6","-12","-8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-2","-16","-8","0"],["0","-1","-6","12","8"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","-48","-768","-2240","16640","90880","96256","35840","4096","0","0"],["0","0","-64","-1072","-3456","21568","108160","768","-99328","-50176","-6144","0"],["0","0","-28","-560","-2480","9408","56576","-54400","-80640","-2048","13312","2048"],["0","0","-4","-132","-888","1776","16992","-27456","-21120","17664","13824","2048"],["0","0","0","-12","-120","144","2208","-4416","-1152","3840","1536","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-48","-864","-4608","-3840","20736","23040","6144","0","0","0"],["0","0","0","-64","-1152","-5600","2560","50304","-10240","-57856","-24576","-2048","0"],["0","0","0","-28","-576","-3072","3360","39168","-39552","-46080","13824","15360","2048"],["0","0","0","-4","-132","-936","1104","15072","-23616","-15744","19200","13824","2048"],["0","0","0","0","-12","-120","144","2208","-4416","-1152","3840","1536","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","-512","-12288","-104448","-360448","-417792","-196608","-32768","0","0","0"],["0","0","0","-768","-16896","-119808","-227328","454656","958464","540672","98304","0","0"],["0","0","0","-384","-7680","-43008","6144","454656","24576","-688128","-491520","-98304","0"],["0","0","0","-64","-1152","-4608","15360","73728","-147456","-122880","147456","147456","32768"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","2304","50688","359424","681984","-1363968","-2875392","-1622016","-294912","0","0","0"],["0","0","0","7680","162048","1046016","1195008","-6592512","-5492736","3219456","3883008","884736","0","0"],["0","0","0","10368","208512","1193472","165888","-10856448","700416","13049856","3833856","-2359296","-884736","0"],["0","0","0","7296","139200","685440","-812544","-8162304","7925760","11231232","-4694016","-5750784","-442368","294912"],["0","0","0","2832","51072","207936","-626688","-3027456","6377472","2930688","-6488064","-2617344","1277952","540672"],["0","0","0","576","9792","31104","-179712","-525312","1990656","-221184","-2433024","0","1032192","294912"],["0","0","0","48","768","1728","-18432","-32256","221184","-129024","-294912","110592","196608","49152"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-768","-19200","-170496","-586752","-227328","2322432","3416064","1720320","294912","0","0","0","0"],["0","0","0","-1792","-44416","-386048","-1242112","-45056","4923392","1196032","-5423104","-4063232","-819200","0","0","0"],["0","0","0","-1664","-40896","-343680","-941568","1195008","6807552","-1130496","-6365184","49152","2473984","688128","0","0"],["0","0","0","-768","-18720","-150144","-302976","1299456","3935232","-4792320","-6746112","1032192","2383872","294912","-98304","0"],["0","0","0","-176","-4256","-32128","-33664","475392","826368","-3022848","-1536000","3452928","2547712","-32768","-360448","-65536"],["0","0","0","-16","-384","-2688","512","57600","36864","-503808","147456","921600","32768","-688128","-393216","-65536"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","384","12032","145408","831488","2166784","1875968","-32768","-851968","-425984","-65536","0","0","0"],["0","0","0","0","704","20736","225024","1019904","1062912","-4300800","-7348224","-2555904","1425408","1114112","196608","0","0"],["0","0","0","0","480","13248","127104","407808","-599040","-4442112","184320","8036352","5529600","49152","-884736","-196608","0"],["0","0","0","0","144","3712","30848","51712","-413952","-1044480","2519040","2703360","-2592768","-3637248","-1015808","131072","65536"],["0","0","0","0","16","384","2688","-512","-57600","-36864","503808","-147456","-921600","-32768","688128","393216","65536"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","1152","25344","184320","450560","233472","1597440","2228224","786432","-98304","-65536","0"],["0","0","0","3264","66432","399616","324608","-2029568","2859008","3432448","-2179072","-2441216","-360448","65536"],["0","0","0","3552","67328","328320","-371712","-3480576","5050368","2813952","-6045696","-3022848","786432","425984"],["0","0","0","1872","33280","128832","-448512","-1884672","4767744","817152","-5210112","-602112","1966080","606208"],["0","0","0","480","8064","24192","-156672","-414720","1769472","-405504","-2211840","221184","1081344","294912"],["0","0","0","48","768","1728","-18432","-32256","221184","-129024","-294912","110592","196608","49152"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-384","-9216","-78080","-268288","-358400","-778240","-2490368","-2523136","-720896","131072","65536","0","0"],["0","0","0","-704","-16704","-133888","-343296","380928","1726464","-786432","319488","3391488","2080768","196608","-65536","0"],["0","0","0","-480","-11360","-86912","-149376","807936","1981440","-3158016","-2838528","1351680","565248","-622592","-229376","0"],["0","0","0","-144","-3424","-25088","-20608","386304","574464","-2519040","-823296","3035136","1646592","-458752","-425984","-65536"],["0","0","0","-16","-384","-2688","512","57600","36864","-503808","147456","921600","32768","-688128","-393216","-65536"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","192","5632","60672","270336","251904","-1228800","-1892352","-393216","638976","393216","65536","0","0"],["0","0","0","0","256","7040","67072","210432","-344064","-2347008","344064","4374528","2555904","-425984","-655360","-131072","0"],["0","0","0","0","112","2880","23808","38656","-324864","-792576","2015232","1990656","-2174976","-2736128","-589824","196608","65536"],["0","0","0","0","16","384","2688","-512","-57600","-36864","503808","-147456","-921600","-32768","688128","393216","65536"]]},"deriv_order":2}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","64","1024","4608","4096","1024","0","0"],["0","0","64","896","2560","-5120","-7168","-2048","0"],["0","0","16","192","192","-2560","768","3072","1024"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"statement":"w(n) E = sum_nu H_nu S^nu L' (with any remainder removed by row annihilators of the parent), where L' is the parent rule at P'_n = P_(n+0) (valid from base 0) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = f(n, theta) = 1 + (1) theta; so the candidate rows satisfy the input rule at every base n >= 0 (the exceptional bases checked exactly), and agreement on the 1 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"from_index":1,"justification":"the parent theorem (LiuWangDerivativeLag) proves P_(m-1) strictly interlacing P_m for every m >= 1; T = f(n, theta) = 1 + (1) theta 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","kind":"weak","orientation":"unordered","status":"claimed"},"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"euler_polynomial_roots_nonpositive","statement":"the polynomial in theta is real-rooted with every root <= 0 for every n >= n0","status":{"kind":"verified"}},{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}},{"evidence":"delta(n) != 0 for every integer n in [0, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 0 + u have one sign)","name":"operator_coefficients_defined","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"0","last":null}},"decision":"holds"},"statement":"the operator coefficients are finite at every n >= n0 (their denominators do not vanish)","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":0,"input_interlacing":{"interlacing":{"from_index":1,"justification":{"premises":["the parent theorem (LiuWangDerivativeLag) proves P_(m-1) strictly interlacing P_m for every m >= 1; T = f(n, theta) = 1 + (1) theta 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":"f(n, theta) = 1 + (1) theta","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (1) theta, for every n >= 0"},"parent_proof":{"route":"liu_wang_derivative_lag","statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]","input_root_region":{"upper":{"closed":true,"value":"0"}},"parent_hypothesis":"every parent row used is zero or real-rooted with all roots in (-inf, 0] (coefficients of one sign)","parent_root_region":{"upper":{"closed":true,"value":"0"}},"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n"},"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,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":true,"statement":"every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 1, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 2"},"hypotheses":[{"evidence":"checked exactly","name":"seed_roots_negative","statement":"P_0 and P_1 are nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"degrees 0 and 1, leading coefficients of one sign","name":"seed_degree_step","statement":"deg P_1 = deg P_0 + 1, with leading coefficients of one sign","status":{"kind":"verified"}},{"evidence":"Bezout test, exact","name":"seed_strict_interlacing","statement":"P_0 strictly interlaces P_1 (simple real roots, no common root)","status":{"kind":"verified"}},{"evidence":"lambda_n = 2 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of lambda_n = 2 at n = 2 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["2"]},"numerator":["2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] A_n + (deg P_(n-1)) [x^2] V_n > 0 for every n >= 2 (deg P_n = deg P_(n-1) + 1 with leading coefficients of one sign)","status":{"kind":"verified"}},{"evidence":"[x^0] c_1 and [x^0] c_2 are zero polynomials","name":"multipliers_vanish_at_zero","statement":"V_n(0) = 0 and W_n(0) = 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"A_n(0) = 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of A_n(0) = 1 at n = 2 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","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) > 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"V_n = 0","name":"derivative_coefficient_nonpositive","statement":"V_n(x) <= 0 for every x < 0 and every n >= 2","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 2 + u have one sign)","name":"lag_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"W_n(x) <= 0 for every x < 0 and every n >= 2","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 2 + u have one sign)","name":"derivative_or_lag_coefficient_negative","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"for every n >= 2 and x < 0, V_n(x) < 0 or W_n(x) < 0 (V_n < 0 on (-inf, 0) for every n, or W_n < 0 there for every n)","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"lean_replay":"not_attempted","lean_route":"sign","normal_form":"P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = (1) + (2) x, V_n(x) = 0, W_n(x) = (1) x","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"liu_wang_derivative_lag"},"parent_proof":{"attempts":[{"detail":"the rule is not P_(n+k) = c(x) P_n with c free of n","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"no theorem matched (weighted_renewal_window: support must satisfy offset = 1 + ell * t_degree; fixed_delay_predecessor_window: more than one weighted lag)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: c_2(n, x) = gamma(n) w(x) with w neither constant nor a constant times a square (x + v_0)^2","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = (1) + (2) x, V_n(x) = 0, W_n(x) = (1) x: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 1, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 2","route":"liu_wang_derivative_lag","success":true,"variant":"liu_wang_derivative_lag"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = (1) + (2) x, V_n(x) = 0, W_n(x) = (1) x","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 1, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 2"],"theorem":"liu_wang_derivative_lag: L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"liu_wang_derivative_lag","rows_checked":10,"statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (1) theta for every n >= 0, the side conditions hold for every n, and every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]"},"kind":"preserver_image"}