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a72760f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | 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
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