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poly.agda
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poly.agda
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open import Algebra
open import Agda.Primitive
open import Data.Nat using (ℕ)
open import Data.List hiding (sum)
sum : {c ℓ : Level} {A : CommutativeRing c ℓ} → List (CommutativeRing.Carrier A) → CommutativeRing.Carrier A
sum {c} {ℓ} {A} [] = 0# where open CommutativeRing A
sum {c} {ℓ} {A} (x ∷ xs) = x + sum xs where open CommutativeRing A
Powerseries : {c ℓ : Level} → CommutativeRing c ℓ → CommutativeRing c ℓ
Powerseries A = record
{ Carrier = ℕ → CommutativeRing.Carrier A
; _≈_ = λ f g → (n : ℕ) → f n ≈ g n
; _+_ = λ f g n → f n + g n
; _*_ = λ f g n → sum (map (λ i → f i * g (n Data.Nat.∸ i)) (upTo (ℕ.suc n)))
; -_ = λ f n → - (f n)
; 0# = λ n → 0#
; 1# = λ { ℕ.zero → 1# ; (ℕ.suc n) → 0# }
; isCommutativeRing = record
{ isRing = record
{ +-isAbelianGroup = record
{ isGroup = record
{ isMonoid = record
{ isSemigroup = record
{ isEquivalence = {!!}
; assoc = λ f g h n → Algebra.Semigroup.assoc (Algebra.Monoid.semigroup (Algebra.Group.monoid (Algebra.AbelianGroup.group (Algebra.Ring.+-abelianGroup (Algebra.CommutativeRing.ring A))))) (f n) (g n) (h n)
; ∙-cong = {!!} }
; identity = {!!} }
; inverse = {!!}
; ⁻¹-cong = {!!} }
; comm = {!!}
}
; *-isMonoid = {!!}
; distrib = {!!}
}
; *-comm = {!!}
}
}
where
open CommutativeRing A