{"chains":[{"edge":{"id":"e-a162795980dd8c54","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-c4e1c7f7d91f2e30","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"87380c2474cfcd7395358977f63a62dd1ed98bbbb3a1dfcc185cf64b722fcad4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f287bed89e07609f","role":"sequence","name":"A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":false,"value":"-1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-5, -1)"}]},"premises":[{"premise":{"position":0,"node_id":"n-f287bed89e07609f","role":"sequence","name":"A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":false,"value":"-1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-5, -1)"},"chains":[{"edge":{"id":"e-c7df247a48c8552e","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-f287bed89e07609f","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":"10f6373de245482d0dca6ec5342374efbffa58c936fa09f9e9ee3c227a8cfcd8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":false,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}]}]}],"truncated":false,"expanded":2}