| import Mathlib | |
| namespace OAI | |
| namespace PiExponent.AddEquivSignedSum | |
| open scoped BigOperators | |
| theorem map_signed_sum {A B : Type*} [AddCommGroup A] [AddCommGroup B] | |
| (E : A ≃+ B) (q : ℕ) (f : Fin (q+2) → A) : | |
| E (∑ k, (-1 : ℤ)^k.val • f k) = ∑ k, (-1 : ℤ)^k.val • E (f k) := by | |
| simp only [map_sum, map_zsmul] | |
| theorem map_signed_sum_eq {A B : Type*} [AddCommGroup A] [AddCommGroup B] | |
| (E : A ≃+ B) (q : ℕ) (f : Fin (q+2) → A) (g : Fin (q+2) → B) | |
| (h : ∀ k, E (f k) = g k) : | |
| E (∑ k, (-1 : ℤ)^k.val • f k) = ∑ k, (-1 : ℤ)^k.val • g k := by | |
| rw [map_signed_sum] | |
| exact Finset.sum_congr rfl (fun k _ => congrArg (fun z => (-1 : ℤ)^k.val • z) (h k)) | |
| end PiExponent.AddEquivSignedSum | |
| end OAI | |