{"chains":[{"edge":{"id":"e-23dacb32496f8efb","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-73aed6c622a33eaf","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":"548588f9c7d56bb9bd6e394466b7717da1daa9c85a6508428e93d8f59b3a2779","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c8b3f39d03de798f","role":"sequence","name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-c8b3f39d03de798f","role":"sequence","name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-01e2cd957653d3da","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-c8b3f39d03de798f","premise_count":1,"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"25782cdb902ac20eeeddad9c4d6d32feaf98867304ed56721e09fbe731948e4f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-811c78acf96cbc19","role":"sequence","name":"Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-811c78acf96cbc19","role":"sequence","name":"Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-b176436450506e78","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-811c78acf96cbc19","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"certificate_hash":"37744ea67cbdd4d249bdfb81088e8abac8de12fa6703f3565d5c289098b0cbcd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]},{"edge":{"id":"e-73f1cbc14f4fad08","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-811c78acf96cbc19","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":"87c843c01db13305c6dd6a773bb85ce41f161557ada85a819199b7b9c147d962","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-811c78acf96cbc19","role":"sequence","name":"Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-811c78acf96cbc19","role":"sequence","name":"Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-b176436450506e78","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-811c78acf96cbc19","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"certificate_hash":"37744ea67cbdd4d249bdfb81088e8abac8de12fa6703f3565d5c289098b0cbcd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},"repeat":true,"premises":[]},{"edge":{"id":"e-796ce5140a0b4b04","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-811c78acf96cbc19","premise_count":1,"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":"58f92b11de764a3c912e390668290bd798639302d0d8c5a313009817373ce6f5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1564556493/268435456"},"upper":{"closed":true,"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]"},"chains":[]}]},{"edge":{"id":"e-b8e0a0fa3ba7bf57","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-811c78acf96cbc19","premise_count":1,"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":"81996fee9e69515ee61ae8c647337f01188d9a23b435f730d4065f7bceac7e37","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]"},"chains":[]}]}]}]},{"edge":{"id":"e-796ce5140a0b4b04","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-811c78acf96cbc19","premise_count":1,"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":"58f92b11de764a3c912e390668290bd798639302d0d8c5a313009817373ce6f5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1564556493/268435456"},"upper":{"closed":true,"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]"}]},"repeat":true,"premises":[]},{"edge":{"id":"e-b8e0a0fa3ba7bf57","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-811c78acf96cbc19","premise_count":1,"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":"81996fee9e69515ee61ae8c647337f01188d9a23b435f730d4065f7bceac7e37","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]"}]},"repeat":true,"premises":[]}]}]},{"edge":{"id":"e-515982525d76b9fb","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-c8b3f39d03de798f","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":"6edf9fb08ae3ef17e5ec90b4ef2ab94c75ec352cd6f0db8e52f97a5f03f572c6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-392f49b496f79fbf","role":"sequence","name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-392f49b496f79fbf","role":"sequence","name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-24564bef34b2c2e5","operation":{"index_shift":-1,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift -1","conclusion_id":"n-392f49b496f79fbf","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":"c1cee09de8e985f0cd2ba88d7a121766a68b49a062d8d65fc827de61301ce16d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-75ccd0453a9b6cec","role":"sequence","name":"Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","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-75ccd0453a9b6cec","role":"sequence","name":"Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","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-0801036cdfff24d2","operation":{"operation":"parent_provers","route":"paper_theorem"},"operation_detail":"paper_theorem","conclusion_id":"n-75ccd0453a9b6cec","premise_count":0,"theorem":"polytool parent provers, route paper_theorem: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"9af6362dca57db0bac2b05eeb707c3159ad751c418442ca87fdfdcc0f9446b79","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":"previous_interlaces_next"}],"premises":[]},"premises":[]},{"edge":{"id":"e-0d3631d0f359c9ce","operation":{"operation":"parent_provers","route":"paper_theorem"},"operation_detail":"paper_theorem","conclusion_id":"n-75ccd0453a9b6cec","premise_count":0,"theorem":"polytool parent provers, route paper_theorem: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"7246e07017856dd0042ba75dd9b9c3f0c39c1a8ece66c8ce940c365457cda5d3","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":"previous_interlaces_next"}],"premises":[]},"premises":[]},{"edge":{"id":"e-a039cba7823b0b25","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-75ccd0453a9b6cec","premise_count":1,"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"d44d40049b95fd6a63c9afcfa609e9d7646a85f365045601067e1fdfc17c39a4","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-e00a1cb428776eaa","role":"sequence","name":"delannoy","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-e00a1cb428776eaa","role":"sequence","name":"delannoy","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-15c233e4385971d1","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-e00a1cb428776eaa","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"certificate_hash":"9c7f633caa7196473351d2b4425cd9ee8d43c2724f2adad5e634b2cdc5aac9af","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-266d85ca03a298fa","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-e00a1cb428776eaa","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":"31ed07789c25145a6fbbd98d0db4108112d8e910e6216a30eb4a673578dc8322","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-e00a1cb428776eaa","role":"sequence","name":"delannoy","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-e00a1cb428776eaa","role":"sequence","name":"delannoy","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-15c233e4385971d1","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-e00a1cb428776eaa","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"certificate_hash":"9c7f633caa7196473351d2b4425cd9ee8d43c2724f2adad5e634b2cdc5aac9af","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":[]},"repeat":true,"premises":[]},{"edge":{"id":"e-e771e35dc9ee3fbd","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-e00a1cb428776eaa","premise_count":1,"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":"d22bd468bd2cd743d01a3ba232cd60c2061a3c8beebb38040f5e5f7e5ccb0908","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1564556493/268435456"},"upper":{"closed":true,"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-6fea5aff4f44fa54","role":"sequence","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-6fea5aff4f44fa54","role":"sequence","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]"},"chains":[]}]},{"edge":{"id":"e-33dba7e79fe9caa8","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-e00a1cb428776eaa","premise_count":1,"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":"efd569ebbe7b8719c045da9b1b2f9612dcbf2177429b29c9c4a3a11adc84c14b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-6fea5aff4f44fa54","role":"sequence","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-6fea5aff4f44fa54","role":"sequence","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]"},"chains":[]}]}]}]},{"edge":{"id":"e-e771e35dc9ee3fbd","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-e00a1cb428776eaa","premise_count":1,"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":"d22bd468bd2cd743d01a3ba232cd60c2061a3c8beebb38040f5e5f7e5ccb0908","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1564556493/268435456"},"upper":{"closed":true,"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-6fea5aff4f44fa54","role":"sequence","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]"}]},"repeat":true,"premises":[]},{"edge":{"id":"e-33dba7e79fe9caa8","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-e00a1cb428776eaa","premise_count":1,"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":"efd569ebbe7b8719c045da9b1b2f9612dcbf2177429b29c9c4a3a11adc84c14b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-6fea5aff4f44fa54","role":"sequence","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]"}]},"repeat":true,"premises":[]}]}]}]}]},{"edge":{"id":"e-fcd19baf8f835cf0","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-392f49b496f79fbf","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":"d5d9576b47c874c4596b3ce911027452c4f2b3f9ea61cdc7b8376e6d8fded954","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-500943bf802dcc95","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 1]"}]},"premises":[{"premise":{"position":0,"node_id":"n-500943bf802dcc95","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 1]"},"chains":[{"edge":{"id":"e-1365676e8cb0fd13","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-2\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-500943bf802dcc95","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":"a9e284e161e9b1783afcf035cd8f19e3090b33322a69654df29dad55c176d366","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":true,"value":"1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c72653d9ab0ee8be","role":"sequence","name":"Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":true,"value":"-1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]"}]},"premises":[{"premise":{"position":0,"node_id":"n-c72653d9ab0ee8be","role":"sequence","name":"Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":true,"value":"-1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]"},"chains":[{"edge":{"id":"e-32a5b77539bbfdf5","operation":{"operation":"theorem_check","theorem":"ma_wang_interval"},"operation_detail":"ma_wang_interval","conclusion_id":"n-c72653d9ab0ee8be","premise_count":0,"theorem":"S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction","lean_route":null,"lean_declarations":["RealRooted.affineEndpointStep_closed","RealRooted.affineEndpointStep_roots_mem_Icc_iff","RealRooted.affineEndpointStep_rootMultiplicity_zero","RealRooted.affineEndpointStep_rootMultiplicity_neg_one","RealRooted.MaWang.endpointStep","RealRooted.MaWang.affineEndpointSequence","RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff"],"certificate_hash":"ef53fced788cfcba2520f5597fbad4a3da33744deafae44fe24823bf42fe0222","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}]}]},{"edge":{"id":"e-3c4e27dedb9c54a8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-500943bf802dcc95","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":"f3ecc14e84a19c598a911339469e2ba68b2103048575d8f0ed92c1d2f4509003","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":true,"value":"1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-392f49b496f79fbf","role":"sequence","name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-392f49b496f79fbf","role":"sequence","name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0]"},"chains":[{"edge":{"id":"e-0178a3d1cd265fb6","operation":{"operation":"theorem_check","theorem":"ma_wang_interval"},"operation_detail":"ma_wang_interval","conclusion_id":"n-392f49b496f79fbf","premise_count":0,"theorem":"S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction","lean_route":null,"lean_declarations":["RealRooted.affineEndpointStep_closed","RealRooted.affineEndpointStep_roots_mem_Icc_iff","RealRooted.affineEndpointStep_rootMultiplicity_zero","RealRooted.affineEndpointStep_rootMultiplicity_neg_one","RealRooted.MaWang.endpointStep","RealRooted.MaWang.affineEndpointSequence","RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff"],"certificate_hash":"3693a8529cdd4207c93b80e3d7c9b1b6eca3d3cd48e6a136f374bba18e7adce0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}]}]}]}]},{"edge":{"id":"e-fa68694d112454ed","operation":{"map":"A026729","operation":"basis_map"},"operation_detail":"A026729","conclusion_id":"n-392f49b496f79fbf","premise_count":1,"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":"9453defebbe177b08bd7fa4ba9956ed37ac1393e3453938a3ba86b38514eb745","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-4"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]"},"chains":[]}]},{"edge":{"id":"e-0178a3d1cd265fb6","operation":{"operation":"theorem_check","theorem":"ma_wang_interval"},"operation_detail":"ma_wang_interval","conclusion_id":"n-392f49b496f79fbf","premise_count":0,"theorem":"S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction","lean_route":null,"lean_declarations":["RealRooted.affineEndpointStep_closed","RealRooted.affineEndpointStep_roots_mem_Icc_iff","RealRooted.affineEndpointStep_rootMultiplicity_zero","RealRooted.affineEndpointStep_rootMultiplicity_neg_one","RealRooted.MaWang.endpointStep","RealRooted.MaWang.affineEndpointSequence","RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff"],"certificate_hash":"3693a8529cdd4207c93b80e3d7c9b1b6eca3d3cd48e6a136f374bba18e7adce0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"repeat":true,"premises":[]}]}]},{"edge":{"id":"e-cf6c2292b090b00e","operation":{"operation":"parent_provers","route":"paper_theorem"},"operation_detail":"paper_theorem","conclusion_id":"n-c8b3f39d03de798f","premise_count":0,"theorem":"polytool parent provers, route paper_theorem: every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"814c4ca297798907dfc9f319136fe7fd9af7893ae816aa0fe168eef86b54b6db","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}]}]}],"truncated":false,"expanded":22}