| import Mathlib.Analysis.Complex.Basic | |
| import Mathlib.Algebra.Module.LinearMap.Defs | |
| import Mathlib.Tactic.NoncommRing | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators | |
| namespace Nonhyperlinear.Diffuseness | |
| /-- Cyclicity equates every polynomial defect moment; no spectral assumption is used. -/ | |
| theorem trace_defect_pow {A:Type*} [Ring A] [Algebra ℂ A] | |
| (f:A→ₗ[ℂ]ℂ) (hf:∀a b,f (a*b)=f (b*a)) (a b:A) (n:ℕ) : | |
| f ((1-a*b)^n)=f ((1-b*a)^n) := by | |
| have hsem : SemiconjBy b (1-a*b) (1-b*a) := by | |
| change b*(1-a*b)=(1-b*a)*b | |
| noncomm_ring | |
| induction n with | |
| | zero => simp | |
| | succ n ih => | |
| have hp:= (hsem.pow_right n).eq | |
| have he : (1-a*b)^(n+1)=(1-a*b)^n-a*(b*(1-a*b)^n) := by | |
| rw [pow_succ'] | |
| noncomm_ring | |
| have he' : (1-b*a)^(n+1)=(1-b*a)^n-((1-b*a)^n*b)*a := by | |
| rw [pow_succ] | |
| noncomm_ring | |
| rw [he,he',map_sub,map_sub,ih,hp,hf a] | |
| end Nonhyperlinear.Diffuseness | |
| end | |
| end OAI | |