{"chains":[{"edge":{"id":"e-5bbae1ad83a2b739","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-1cc5242f219c3841","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"f94c748c1336ae12c99644628e53ef74fbcc11fc552263e8bd838d9492022380","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-3ffff24fcb4f40a0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","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-3ffff24fcb4f40a0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","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-9baf8deff42ffdae","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-3ffff24fcb4f40a0","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":"halfGrowth","lean_declarations":["RealRooted.derivLag_strictInterl_of_nonnegCoeffs","RealRooted.derivLag_strictInterl_and_noCommonRoot","RealRooted.derivLag_eval_zero_pos"],"certificate_hash":"447d3e61ec8dd9e298d858fbb4b916ce19bdd0e612e5095f4b4d11a30fadbca9","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":2}