{"chains":[{"edge":{"id":"e-216cbdf61e3ed56f","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-1e2f1b7f566b371c","premise_count":0,"theorem":"affine current multiplier with an affine initial row (module documentation, \"The affine-current theorem\"): with beta = b/d, K = beta^2 + a beta - c > 0 and eta = m/d >= 0, every real polynomial with all roots in E = [-beta - sqrt(K), -beta + sqrt(K)] has a real-rooted image, and no interval of positive length outside E works (Theorem 1); for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"0a6de4c4cd055cfc120b210d9cec81478518f044990b50b6d719f4e0ea460845","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"[-7 - sqrt(56), -7 + sqrt(56)]","output_interval":"(-inf, +inf)","statement":"every real polynomial with all roots in E = [-7 - sqrt(56), -7 + sqrt(56)] has a real-rooted image"},{"claim":"basis_map_preserver","input_interval":"[-7 - sqrt(56), -7 + sqrt(56)]","output_interval":"[-15 - 2 sqrt(56), -15 + 2 sqrt(56)]","statement":"every real polynomial with all roots in E = [-7 - sqrt(56), -7 + sqrt(56)] has an image with all roots in I = [-15 - 2 sqrt(56), -15 + 2 sqrt(56)]"},{"claim":"basis_map_preserver","input_interval":"[-7 - sqrt(56), 0]","output_interval":"[-15 - 2 sqrt(56), -15 + 2 sqrt(56)]","statement":"every PF input with all roots in [-7 - sqrt(56), 0] has a PF image; no larger [-R, 0] works"}],"premises":[]},"premises":[]},{"edge":{"id":"e-8666650c0d0b3b54","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-1e2f1b7f566b371c","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"61f0ae813a7583c66165837c9d70f0a810775110ded84258b55db32bca238759","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},"premises":[]},{"edge":{"id":"e-7d90ede2da8763e3","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-1e2f1b7f566b371c","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":"1090e974b89a6db6a15dbbd3229b7c3e410a4defc4e412f257cb278e604d17a1","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-1e2f1b7f566b371c","role":"sequence","name":"Pascal-(1,7,1) array.","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-1e2f1b7f566b371c","role":"sequence","name":"Pascal-(1,7,1) array.","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-a31f5c33917b15a7","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-1e2f1b7f566b371c","premise_count":0,"theorem":"L. 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