{"chains":[{"edge":{"id":"e-a526bc158f95a9e3","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-ff69dadb0811f5e6","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":"9b55553c1ca89452e512ff125a94bdbe438a3d2e9f523b6e9f6102352d2ddaf4","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":"strict","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]},{"edge":{"id":"e-ea394865db18a818","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-ff69dadb0811f5e6","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":"c730eae8e0ec70e4863e53cc9878f9393629e2e85b2cab5460c006fa5465e85c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}],"truncated":false,"expanded":2}
