{"chains":[{"edge":{"id":"e-82339c54d84ff82b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-94d1bd0c0d842b87","premise_count":1,"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","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"b544a79f098257bf81340c99d8538a0af65c1904d95d8cb2b7e900feebca8405","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"-1/2"}}}],"premises":[{"position":0,"node_id":"n-317113238d23620b","role":"sequence","name":"basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-9"},"upper":{"closed":false,"value":"-1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-9, -1)"}]},"premises":[{"premise":{"position":0,"node_id":"n-317113238d23620b","role":"sequence","name":"basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-9"},"upper":{"closed":false,"value":"-1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-9, -1)"},"chains":[{"edge":{"id":"e-c4464c8cda467541","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-317113238d23620b","premise_count":1,"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","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"4a6b3e865e24e3287409cc7829bc766608dae9ee73b65f09a323c6ab8cfeb2a6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-9"},"upper":{"closed":false,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6466a8550c08a52f","role":"sequence","name":"Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-8"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0)"}]},"premises":[{"premise":{"position":0,"node_id":"n-6466a8550c08a52f","role":"sequence","name":"Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-8"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0)"},"chains":[{"edge":{"id":"e-7bedda0241d23786","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-6466a8550c08a52f","premise_count":0,"theorem":"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","lean_route":null,"lean_declarations":[],"certificate_hash":"3604ba1a60d40762b516575aa755b54f8fe74b5cef3fb8b27457bff0362de4e9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-8"},"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}]}]},{"edge":{"id":"e-f1629c32ad3d0109","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-317113238d23620b","premise_count":1,"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","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"450f8c9a8d29ae3f229f0cde5ceab149ff6c0c0c67b8e9099ad3c88eac7604b8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-9"},"upper":{"closed":false,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6466a8550c08a52f","role":"sequence","name":"Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-8"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0)"}]},"premises":[{"premise":{"position":0,"node_id":"n-6466a8550c08a52f","role":"sequence","name":"Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-8"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0)"},"chains":[{"edge":{"id":"e-7bedda0241d23786","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-6466a8550c08a52f","premise_count":0,"theorem":"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","lean_route":null,"lean_declarations":[],"certificate_hash":"3604ba1a60d40762b516575aa755b54f8fe74b5cef3fb8b27457bff0362de4e9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-8"},"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},"repeat":true,"premises":[]}]}]}]}]}],"truncated":false,"expanded":4}