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for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"87d2d0f1cb7df9a7ab60926087852fafddc87229b9045cdf3cae14ea01948542","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-7"},"upper":{"closed":false,"value":"-3"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-30cb3cad216ed189","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-6"},"upper":{"closed":false,"value":"-2"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-6, -2)"}]},"repeat":true,"premises":[]},{"edge":{"id":"e-785b355515669be0","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c2b7c74f8491f01b","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":"da97cbfc907ffe2ceb5fd9ec077d60e6b83fa0d1fdd47e0d9341c76500f95afa","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":"unordered"}],"premises":[{"position":0,"node_id":"n-30cb3cad216ed189","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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]"}]},"repeat":true,"premises":[]},{"edge":{"id":"e-dddb678c4dbb8562","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c2b7c74f8491f01b","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":"1be534597a4f31b4faab5ae64c14e1f2c75cde5d77a62871c2f4b0c3c66b5f1c","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":"unordered"}],"premises":[{"position":0,"node_id":"n-30cb3cad216ed189","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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]"}]},"repeat":true,"premises":[]}]}]}]}]}],"truncated":false,"expanded":33}