{"id":"e-9c6e37fd70f39446","operation":{"index_shift":-1,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-1\"}"},"operation_detail":"affine, shift -1","conclusion_id":"n-f0c4b6406e0da4a3","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":"7f81dc997268c8d25cadc1232f018af6480b41ff66fe1dda376b115c8b6cbe54","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":12753,"certificate_compressed_size":2697,"conclusion_name":"Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n.","claims":[{"claim":"real_rooted","from_index":1,"region":{}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b0b0e9fb10acb2e2","role":"sequence","name":"hermite_he","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}],"cited_by":[{"node_id":"n-f0c4b6406e0da4a3","claim":"real_rooted"},{"node_id":"n-f0c4b6406e0da4a3","claim":"interlacing"}]}