Created
October 18, 2023 10:34
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Monoid Sum
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open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong) | |
record Monoid (m : Set) : Set where | |
infixl 6 _⊕_ | |
field | |
_⊕_ : m → m → m | |
associativity : ∀{a b c} → ((a ⊕ b) ⊕ c) ≡ (a ⊕ (b ⊕ c)) | |
open Monoid {{...}} | |
data NList (m : Set) : Set where | |
Sing : m → NList m | |
Cons : m → NList m → NList m | |
infixr 5 _++_ | |
_++_ : ∀ {m} → NList m → NList m → NList m | |
Sing x ++ ys = Cons x ys | |
Cons x xs ++ ys = Cons x (xs ++ ys) | |
data Bin (m : Set) : Set where | |
Leaf : m → Bin m | |
Plus : Bin m → Bin m → Bin m | |
bin2nlist : ∀ {m} → Bin m → NList m | |
bin2nlist (Leaf m) = Sing m | |
bin2nlist (Plus b₁ b₂) = bin2nlist b₁ ++ bin2nlist b₂ | |
sumNList : ∀ {m}{{M : Monoid m}} → NList m → m | |
sumNList {_} (Sing m) = m | |
sumNList {m} (Cons x xs) = x ⊕ sumNList xs | |
sumBin : ∀ {m}{{M : Monoid m}} → Bin m → m | |
sumBin (Leaf m) = m | |
sumBin (Plus b₁ b₂) = sumBin b₁ ⊕ sumBin b₂ | |
sum-dist : ∀{m} {{M : Monoid m}} → (xs ys : NList m) → sumNList (xs ++ ys) ≡ sumNList xs ⊕ sumNList ys | |
sum-dist {_} (Sing x) ys = refl | |
sum-dist {m} (Cons x xs) ys rewrite associativity {m} {x} {sumNList xs} {sumNList ys} = | |
cong (_⊕_ x) (sum-dist xs ys) | |
sum-universal : ∀{m} {{M : Monoid m}} → (b : Bin m) → sumNList (bin2nlist b) ≡ sumBin b | |
sum-universal {m} (Leaf x) = refl | |
sum-universal {m} (Plus b₁ b₂) | |
rewrite sum-dist (bin2nlist b₁) (bin2nlist b₂) | |
rewrite sum-universal b₁ | |
rewrite sum-universal b₂ = refl |
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