{"id":"e-6e459812af5dc78f","operation":{"index_shift":-1,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"-2\",\"intercept\":\"-1\"}"},"operation_detail":"affine, shift -1","conclusion_id":"n-a52b649d7348b7a5","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":"6481b35b66795aae2a298526943ecd4432c88b9759d7a483b0420d146f1149f8","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":10434,"certificate_compressed_size":2943,"conclusion_name":"Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1/2"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4f6e747b794bcc2e","role":"sequence","name":"Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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]"}],"cited_by":[{"node_id":"n-a52b649d7348b7a5","claim":"interlacing"},{"node_id":"n-a52b649d7348b7a5","claim":"real_rooted"}]}