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for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"cfc69058f92569cc3bfec4e4bc7d72fbbea6f16aebd9163490d3756b83cb32ff","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-384c17a53af3442c","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-384c17a53af3442c","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-62a1b0a86f60f435","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-384c17a53af3442c","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"5f6c4d3068c3fffafd2c22fbfcd53bd7f8b9d97c0187b6f1badafbc5175b2587","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a6a633b989da1879","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-a6a633b989da1879","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-00bae21f0fe8326e","operation":{"operation":"theorem_check","theorem":"hypergeometric_row"},"operation_detail":"hypergeometric_row","conclusion_id":"n-a6a633b989da1879","premise_count":0,"theorem":"Row-hypergeometric rows: an exact row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, proved for every n by an invariant row-annihilator system of the rule, writes every row as U_0(n) x^v (or its reversal) times the image of (1 + x)^M(n) under a product of nonnegative multiplier sequences: C^k (C > 0); k -> M'!/(M' - k)!, whose exponential generating function is (1 + x)^M' (Polya--Schur, Theorem 7.1 (3) => (1) of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601: a sequence is a multiplier sequence when G(x) = sum_k lambda_k x^k / k! or G(-x) can be written C x^m e^(a x) prod_k (1 + alpha_k x) with a >= 0, alpha_k >= 0); k -> phi(k) for a polynomial phi with only real roots, all <= 0 (Laguerre's theorem, RealRooted isPFMultiplierSequence_eval_of_roots_nonpos); k -> 1/(alpha)_k for alpha > 0 (RealRooted isPFMultiplierSequence_inv_ascPochhammer, via the generalized Laguerre polynomials); products of PF multiplier sequences are PF multiplier sequences (RealRooted IsPFMultiplierSequence.mul) and map real-rooted polynomials with nonnegative coefficients to such polynomials (IsPFMultiplierSequence.diagonalOperator_isPF)","lean_route":null,"lean_declarations":["RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_inv_ascPochhammer","RealRooted.IsPFMultiplierSequence.mul","RealRooted.IsPFMultiplierSequence.diagonalOperator_isPF"],"certificate_hash":"a47f4943b5a1fe39657839f7186c1dcce4f719d5c1f4097ae64d79451ca4f0d9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}}],"premises":[]},"premises":[]},{"edge":{"id":"e-8a6d49b40ba3d6fa","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-a6a633b989da1879","premise_count":0,"theorem":"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 g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"db5ec03d9e037f9411fe9e61d02749ea332325d747c7bd08219acd9e5211514a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]},{"edge":{"id":"e-7b683ba9a55bfcf0","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-a6a633b989da1879","premise_count":0,"theorem":"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","lean_route":null,"lean_declarations":[],"certificate_hash":"5ba5281ad3a6bfffb2b5128d220881957927716a30aee6c5a736f1a0aafd0d9b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]},{"edge":{"id":"e-cb24020f7808d9ac","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-a6a633b989da1879","premise_count":0,"theorem":"polytool parent provers, route periodic_product: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"9346e1856a97a48048b7936abe125ab6b19705a512d426375653f0d8e823cd8b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},"premises":[]}]}]}]}]}],"truncated":false,"expanded":140}