open import 1Lab.Prelude

open import Data.Fin.Base
open import Data.Sum

module Data.Fin.Closure where

private variable
  ℓ : Level
  A B C : Type ℓ
  k l m n : Nat

Closure of finite sets🔗

In this module, we prove that the finite sets are closed under “typal arithmetic”: The initial and terminal objects are finite (they have 1 and 0 elements respectively), products of finite sets are finite, coproducts of finite sets are finite, and functions between finite sets are finite. Moreover, these operations all correspond to arithmetic operations on the natural number indices: $[n] \uplus [m] = [n + m]$, etc.

Zero, one, successors🔗

The finite set $[0]$ is an initial object, and the finite set $[1]$ is a terminal object:

Finite-zero-is-initial : Fin  ≃ ⊥
Finite-zero-is-initial .fst ()
Finite-zero-is-initial .snd .is-eqv ()

Finite-one-is-contr : is-contr (Fin )
Finite-one-is-contr .centre = fzero
Finite-one-is-contr .paths fzero = refl

The successor operation on indices corresponds to taking coproducts with the unit set.

Finite-successor : Fin (suc n) ≃ (⊤ ⊎ Fin n)
Finite-successor {n} = Iso→Equiv (f , iso g rinv linv) where
  f : Fin (suc n) → ⊤ ⊎ Fin n
  f fzero = inl tt
  f (fsuc x) = inr x

  g : ⊤ ⊎ Fin n → Fin (suc n)
  g (inr x) = fsuc x
  g (inl _) = fzero

  rinv : is-right-inverse g f
  rinv (inr _) = refl
  rinv (inl _) = refl

  linv : is-left-inverse g f
  linv fzero = refl
  linv (fsuc x) = refl

We can also phrase this equivalence in a particularly strong way, which applies to dependent products over finite successor types:

Fin-suc-universal
  : ∀ {ℓ} {n} {A : Fin (suc n) → Type ℓ}
  → (∀ x → A x) ≃ (A fzero × (∀ x → A (fsuc x)))
Fin-suc-universal = Iso→Equiv λ where
  .fst f → f fzero , (λ x → f (fsuc x))

  .snd .is-iso.inv (z , f) fzero    → z
  .snd .is-iso.inv (z , f) (fsuc x) → f x

  .snd .is-iso.rinv x → refl

  .snd .is-iso.linv k i fzero    → k fzero
  .snd .is-iso.linv k i (fsuc n) → k (fsuc n)

Addition🔗

For binary coproducts, we prove the correspondence with addition in steps, to make the proof clearer:

Finite-coproduct : (Fin n ⊎ Fin m) ≃ Fin (n + m)
Finite-coproduct {zero} {m}  =
  (Fin  ⊎ Fin m) ≃⟨ ⊎-apl Finite-zero-is-initial ⟩≃
  (⊥ ⊎ Fin m)     ≃⟨ ⊎-zerol ⟩≃
  Fin m           ≃∎
Finite-coproduct {suc n} {m} =
  (Fin (suc n) ⊎ Fin m) ≃⟨ ⊎-apl Finite-successor ⟩≃
  ((⊤ ⊎ Fin n) ⊎ Fin m) ≃⟨ ⊎-assoc ⟩≃
  (⊤ ⊎ (Fin n ⊎ Fin m)) ≃⟨ ⊎-apr (Finite-coproduct {n} {m}) ⟩≃
  (⊤ ⊎ Fin (n + m))     ≃⟨ Finite-successor e⁻¹ ⟩≃
  Fin (suc (n + m))     ≃∎

Sums🔗

We also have a correspondence between “coproducts” and “addition” in the iterated case: If you have a family of finite types (represented by a map to their cardinalities), the dependent sum of that family is equivalent to the iterated binary sum of the cardinalities:

sum : ∀ n → (Fin n → Nat) → Nat
sum zero f = zero
sum (suc n) f = f fzero + sum n (f ∘ fsuc)

In this case, the isomorphism is constructed directly:

Finite-sum : (B : Fin n → Nat) → Σ (Fin _) (Fin ∘ B) ≃ Fin (sum n B)
Finite-sum {zero} B .fst ()
Finite-sum {zero} B .snd .is-eqv ()
Finite-sum {suc n} B =
  Finite-coproduct .fst ∘ f ,
  ∙-is-equiv (is-iso→is-equiv f-iso) (Finite-coproduct .snd)
    where
      rec = Finite-sum (B ∘ fsuc)
      module rec = Equiv rec

      f : Σ _ (Fin ∘ B) → Fin (B fzero) ⊎ Fin (sum n (B ∘ fsuc))
      f (fzero , x) = inl x
      f (fsuc x , y) = inr (rec .fst (x , y))

      f-iso : is-iso f
      f-iso .is-iso.inv (inl x) = fzero , x
      f-iso .is-iso.inv (inr x) with rec.from x
      ... | x , y = fsuc x , y

      f-iso .is-iso.rinv (inl x) = refl
      f-iso .is-iso.rinv (inr x) = ap inr (rec.ε _)

      f-iso .is-iso.linv (fzero , x) = refl
      f-iso .is-iso.linv (fsuc x , y) =
        Σ-pathp (ap (fsuc ∘ fst) (rec.η _)) (ap snd (rec.η _))

Multiplication🔗

Recall (from middle school) that the product $n \times m$ is the same thing as summing together $n$ copies of the number $m$. Correspondingly, we can use the theorem above for general sums to establish the case of binary products:

Finite-product : (Fin n × Fin m) ≃ Fin (n * m)
Finite-product {n} {m} =
  (Fin n × Fin m)       ≃⟨ Finite-sum (λ _ → m) ⟩≃
  Fin (sum n (λ _ → m)) ≃⟨ cast (sum≡* n m) , cast-is-equiv _ ⟩≃
  Fin (n * m)           ≃∎
  where
    sum≡* : ∀ n m → sum n (λ _ → m) ≡ n * m
    sum≡* zero m = refl
    sum≡* (suc n) m = ap (m +_) (sum≡* n m)