open import 1Lab.Prelude

open import Data.Maybe.Properties
open import Data.Maybe.Base
open import Data.Fin.Base
open import Data.Nat.Base using (s≤s)
open import Data.Dec
open import Data.Irr
open import Data.Sum

open import Meta.Invariant

import Data.Nat.Order as Nat
import Data.Nat.Base as Nat

module Data.Fin.Properties where

Finite sets - properties🔗

Ordering🔗

As noted in Data.Fin.Base, we’ve set up the ordering on Fin so that we can re-use all the proofs about the ordering on Nat.

However, there are still quite a few interesting things one can say about skip and squish. In particular, we can prove the simplicial identities, which characterize the interactions between these two functions.

These lemmas might seem somewhat arbitrary and complicated, which is true! However, they are enough to describe all the possible interactions of skip and squish, which in turn are the building blocks for every monotone function between Fin, so it’s not that surprising that they would be a bit of a mess!

skip-comm : ∀ {n} (i j : Fin (suc n)) → i ≤ j
          → ∀ x → skip (weaken i) (skip j x) ≡ skip (fsuc j) (skip i x)
skip-comm i j le x with fin-view i | fin-view j | le | fin-view x
... | zero  | zero  | _      | _       = refl
... | zero  | suc _ | _      | _       = refl
... | suc i | suc j | le     | zero    = refl
... | suc i | suc j | s≤s le | (suc x) = ap fsuc (skip-comm i j le x)

drop-comm : ∀ {n} (i j : Fin n) → i ≤ j
          → ∀ x → squish j (squish (weaken i) x) ≡ squish i (squish (fsuc j) x)
drop-comm i j le x with fin-view i | fin-view j | le | fin-view x
... | zero  | zero  | le     | zero  = refl
... | zero  | zero  | le     | suc x = refl
... | zero  | suc j | le     | zero  = refl
... | zero  | suc j | le     | suc x = refl
... | suc i | suc j | le     | zero  = refl
... | suc i | suc j | s≤s le | suc x = ap fsuc (drop-comm i j le x)

squish-skip-comm : ∀ {n} (i : Fin (suc n)) (j : Fin n) → i < fsuc j
                 → ∀ x → squish (fsuc j) (skip (weaken i) x) ≡ skip i (squish j x)
squish-skip-comm i j le x with fin-view i | fin-view j | le | fin-view x
... | zero | zero  | s≤s p | zero  = refl
... | zero | zero  | s≤s p | suc _ = refl
... | zero | suc _ | s≤s p | zero  = refl
... | zero | suc _ | s≤s p | suc _ = refl
... | suc i | (suc j) | (Nat.s≤s p) | zero = refl
... | suc i | (suc j) | (Nat.s≤s p) | (suc x) =
  ap fsuc (squish-skip-comm i j p x)

squish-skip : ∀ {n} (i j : Fin n) → i ≡ j
            → ∀ x → squish j (skip (weaken j) x) ≡ x
squish-skip i j p x with fin-view i | fin-view j | fin-view x
... | zero    | zero    | x       = refl
... | zero    | (suc j) | x       = absurd (fzero≠fsuc p)
... | (suc i) | zero    | x       = refl
... | (suc i) | (suc j) | zero    = refl
... | (suc i) | (suc j) | (suc x) =
  ap fsuc (squish-skip i j (fsuc-inj p) x)

squish-skip-fsuc : ∀ {n} (i : Fin (suc n)) (j : Fin n) → i ≡ fsuc j
                 → ∀ x → squish j (skip i x) ≡ x
squish-skip-fsuc i j p x with fin-view i | fin-view j | fin-view x
... | zero | zero | x = refl
... | zero | suc j | x = absurd (fzero≠fsuc p)
... | suc i | suc j | zero  = refl
... | suc i | suc j | suc x = ap fsuc (squish-skip-fsuc i j (fsuc-inj p) x)
... | suc i | zero | x with fin-view i | x
... | zero | zero = refl
... | zero | suc x = refl
... | suc i | zero = refl
... | suc i | suc x = absurd (Nat.zero≠suc λ i → Nat.pred (p (~ i) .lower))

Fin-suc : ∀ {n} → Fin (suc n) ≃ Maybe (Fin n)
Fin-suc = Iso→Equiv (to , iso from ir il) where
  to : ∀ {n} → Fin (suc n) → Maybe (Fin n)
  to i with fin-view i
  ... | suc i = just i
  ... | zero  = nothing

  from : ∀ {n} → Maybe (Fin n) → Fin (suc n)
  from (just x) = fsuc x
  from nothing  = fzero

  ir : is-right-inverse from to
  ir nothing = refl
  ir (just x) = refl

  il : is-left-inverse from to
  il i with fin-view i
  ... | suc i = refl
  ... | zero  = refl

Fin-peel : ∀ {l k} → Fin (suc l) ≃ Fin (suc k) → Fin l ≃ Fin k
Fin-peel {l} {k} sl≃sk = Maybe-injective (Equiv.inverse Fin-suc ∙e sl≃sk ∙e Fin-suc)

Fin-injective : ∀ {l k} → Fin l ≃ Fin k → l ≡ k
Fin-injective {zero} {zero} l≃k = refl
Fin-injective {zero} {suc k} l≃k with equiv→inverse (l≃k .snd) fzero
... | ()
Fin-injective {suc l} {zero} l≃k with l≃k .fst fzero
... | ()
Fin-injective {suc l} {suc k} sl≃sk = ap suc $ Fin-injective (Fin-peel sl≃sk)

avoid-injective
  : ∀ {n} (i : Fin (suc n)) {j k : Fin (suc n)} {i≠j : i ≠ j} {i≠k : i ≠ k}
  → avoid i j i≠j ≡ avoid i k i≠k → j ≡ k
avoid-injective i {j} {k} {i≠j} {i≠k} p with fin-view i | fin-view j | fin-view k
... | zero | zero | _ = absurd (i≠j refl)
... | zero | suc j | zero = absurd (i≠k refl)
... | zero | suc j | suc k = ap fsuc p
... | suc i | zero | zero = refl
avoid-injective {suc n} _ p | suc i | zero  | suc k = absurd (fzero≠fsuc p)
avoid-injective {suc n} _ p | suc i | suc j | zero  = absurd (fsuc≠fzero p)
avoid-injective {suc n} _ p | suc i | suc j | suc k = ap fsuc (avoid-injective {n} i {j} {k} (fsuc-inj p))

skip-injective
  : ∀ {n} (i : Fin (suc n)) (j k : Fin n)
  → skip i j ≡ skip i k → j ≡ k
skip-injective i j k p with fin-view i | fin-view j | fin-view k
... | zero  | j     | k     = fsuc-inj p
... | suc i | zero  | zero  = refl
... | suc i | zero  | suc k = absurd (fzero≠fsuc p)
... | suc i | suc j | zero  = absurd (fsuc≠fzero p)
... | suc i | suc j | suc k = ap fsuc (skip-injective i j k (fsuc-inj p))

skip-skips
  : ∀ {n} (i : Fin (suc n)) (j : Fin n)
  → skip i j ≠ i
skip-skips i j p with fin-view i | fin-view j
... | zero  | j     = fsuc≠fzero p
... | suc i | zero  = fzero≠fsuc p
... | suc i | suc j = skip-skips i j (fsuc-inj p)

avoid-skip
  : ∀ {n} (i : Fin (suc n)) (j : Fin n) {neq : i ≠ skip i j}
  → avoid i (skip i j) neq ≡ j
avoid-skip i j with fin-view i | fin-view j
... | zero  | zero  = refl
... | zero  | suc j = refl
... | suc i | zero  = refl
... | suc i | suc j = ap fsuc (avoid-skip i j)

skip-avoid
  : ∀ {n} (i : Fin (suc n)) (j : Fin (suc n)) {i≠j : i ≠ j}
  → skip i (avoid i j i≠j) ≡ j
skip-avoid i j {i≠j} with fin-view i | fin-view j
... | zero | zero = absurd (i≠j refl)
skip-avoid {suc n} _ _ | zero  | suc j = refl
skip-avoid {suc n} _ _ | suc i | zero  = refl
skip-avoid {suc n} _ _ | suc i | suc j = ap fsuc (skip-avoid i j)

Iterated products and sums🔗

We can break down $Π -types$ and $Σ -types$ over finite sets as iterated products and sums, respectively.

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

  .snd .is-iso.inv (z , s) → fin-cons z s

  .snd .is-iso.rinv x → refl

  .snd .is-iso.linv k i fzero               → k (fin zero ⦃ forget auto ⦄)
  .snd .is-iso.linv k i (fin (suc n) ⦃ b ⦄) → k (fin (suc n) ⦃ b ⦄)

Fin-suc-Σ
  : ∀ {ℓ} {n} {A : Fin (suc n) → Type ℓ}
  → Σ (Fin (suc n)) A ≃ (A fzero ⊎ Σ (Fin n) (A ∘ fsuc))
Fin-suc-Σ {A = A} = Iso→Equiv (to , iso from ir il) where
  to : ∫ₚ A → A fzero ⊎ ∫ₚ (A ∘ fsuc)
  to (i , a) with fin-view i
  ... | zero  = inl a
  ... | suc x = inr (x , a)

  from : A fzero ⊎ ∫ₚ (A ∘ fsuc) → ∫ₚ A
  from (inl x)       = fzero , x
  from (inr (x , a)) = fsuc x , a

  ir : is-right-inverse from to
  ir (inl x) = refl
  ir (inr x) = refl

  il : is-left-inverse from to
  il (i , a) with fin-view i
  ... | zero  = refl
  ... | suc _ = refl

Finite choice🔗

An important fact about the (standard) finite sets in constructive mathematics is that they always support choice, which we phrase below as a “search” operator: if $M$ is any Monoidal functor on types, then it commutes with products. Since $Π -types$ over $[n]$ are $n -ary$ iterated products, we have that $M$ commutes with $Π .$

Fin-Monoidal
  : ∀ {ℓ} n {A : Fin n → Type ℓ} {M}
      (let module M = Effect M)
  → ⦃ Monoidal M ⦄
  → (∀ x → M.₀ (A x)) → M.₀ (∀ x → A x)
Fin-Monoidal zero _ = invmap (λ _ ()) _ munit
Fin-Monoidal (suc n) k =
  Fin-suc-Π e⁻¹ <≃> (k 0 <,> Fin-Monoidal n (k ∘ fsuc))

_ = Idiom

In particular, instantiating $M$ with the propositional truncation (which is an Idiom and hence Monoidal), we get a version of the axiom of choice for finite sets.

finite-choice
  : ∀ {ℓ} n {A : Fin n → Type ℓ}
  → (∀ x → ∥ A x ∥) → ∥ (∀ x → A x) ∥
finite-choice n = Fin-Monoidal n

An immediate consequence is that surjections into a finite set (thus, between finite sets) merely split:

finite-surjection-split
  : ∀ {ℓ} {n} {B : Type ℓ}
  → (f : B → Fin n) → is-surjective f
  → ∥ (∀ x → fibre f x) ∥
finite-surjection-split f = finite-choice _

Dually, we have that any Alternative functor $M$ commutes with $Σ -types$ on finite sets, since those are iterated sums.

Fin-Alternative
  : ∀ {ℓ} n {A : Fin n → Type ℓ} {M}
      (let module M = Effect M)
  → ⦃ Alternative M ⦄
  → (∀ x → M.₀ (A x)) → M.₀ (Σ (Fin n) A)
Fin-Alternative zero _ = invmap (λ ()) (λ ()) empty
Fin-Alternative (suc n) k =
  Fin-suc-Σ e⁻¹ <≃> (k 0 <+> Fin-Alternative n (k ∘ fsuc))

As a consequence, instantiating $M$ with Dec, we get that finite sets are exhaustible and omniscient, which means that any family of decidable types indexed by a finite sets yields decidable $Π -types$ and $Σ -types,$ respectively.

instance
  Dec-Fin-∀
    : ∀ {n ℓ} {A : Fin n → Type ℓ}
    → ⦃ ∀ {x} → Dec (A x) ⦄ → Dec (∀ x → A x)
  Dec-Fin-∀ {n} ⦃ d ⦄ = Fin-Monoidal n (λ _ → d)

  Dec-Fin-Σ
    : ∀ {n ℓ} {A : Fin n → Type ℓ}
    → ⦃ ∀ {x} → Dec (A x) ⦄ → Dec (Σ (Fin n) A)
  Dec-Fin-Σ {n} ⦃ d ⦄ = Fin-Alternative n λ _ → d

Fin-omniscience
  : ∀ {n ℓ} (P : Fin n → Type ℓ) ⦃ _ : ∀ {x} → Dec (P x) ⦄
  → (Σ[ j ∈ Fin n ] P j × ∀ k → P k → j ≤ k) ⊎ (∀ x → ¬ P x)
Fin-omniscience {zero} P = inr λ ()
Fin-omniscience {suc n} P with holds? (P 0)
... | yes here = inl (0 , here , λ _ _ → 0≤x)
... | no ¬here with Fin-omniscience (P ∘ fsuc)
... | inl (ix , pix , least) = inl (fsuc ix , pix , fin-cons (λ here → absurd (¬here here)) λ i pi → Nat.s≤s (least i pi))
... | inr nowhere = inr (fin-cons ¬here nowhere)

Fin-omniscience-neg
  : ∀ {n ℓ} (P : Fin n → Type ℓ) ⦃ _ : ∀ {x} → Dec (P x) ⦄
  → (∀ x → P x) ⊎ (Σ[ j ∈ Fin n ] ¬ P j × ∀ k → ¬ P k → j ≤ k)
Fin-omniscience-neg P with Fin-omniscience (¬_ ∘ P)
... | inr p = inl λ i → dec→dne (p i)
... | inl (j , ¬pj , least) = inr (j , ¬pj , least)

Fin-find
  : ∀ {n ℓ} {P : Fin n → Type ℓ} ⦃ _ : ∀ {x} → Dec (P x) ⦄
  → ¬ (∀ x → P x)
  → Σ[ x ∈ Fin n ] ¬ P x × ∀ y → ¬ P y → x ≤ y
Fin-find {P = P} ¬p with Fin-omniscience-neg P
... | inl p = absurd (¬p p)
... | inr p = p

Injections and surjections🔗

The standard finite sets are Dedekind-finite, which means that every injection $[n] ↪ [n]$ is a bijection. We prove this by a straightforward but annoying induction on $n .$

Fin-injection→equiv
  : ∀ {n} (f : Fin n → Fin n)
  → injective f → is-equiv f
Fin-injection→equiv {zero} f inj .is-eqv ()
Fin-injection→equiv {suc n} f inj .is-eqv i with f 0 ≡? i
... | yes p = contr (0 , p) λ (j , p') → Σ-prop-path! (inj (p ∙ sym p'))
... | no ¬p = contr fib cen where
  rec = Fin-injection→equiv {n}
    (λ x → avoid (f 0) (f (fsuc x)) (Nat.zero≠suc ∘ ap lower ∘ inj))
    (λ p → fsuc-inj (inj (avoid-injective (f 0) p)))
    .is-eqv (avoid (f 0) i ¬p)

  fib : fibre f i
  fib = fsuc (rec .centre .fst) , avoid-injective (f 0) (rec .centre .snd)

  cen : ∀ x → fib ≡ x
  cen (i , p) with fin-view i
  ... | zero  = absurd (¬p p)
  ... | suc j = Σ-prop-path! (ap (fsuc ∘ fst)
      (rec .paths (j , ap₂ (avoid (f 0)) p prop!)))

Since every surjection between finite sets splits, any surjection $f : [n] ↠ [n]$ has an injective right inverse, which is thus a bijection; by general properties of equivalences, this implies that $f$ is also a bijection.

Fin-surjection→equiv
  : ∀ {n} (f : Fin n → Fin n)
  → is-surjective f → is-equiv f
Fin-surjection→equiv f surj = case finite-surjection-split f surj of λ split →
  left-inverse→equiv (snd ∘ split)
    (Fin-injection→equiv (fst ∘ split)
      (right-inverse→injective f (snd ∘ split)))

Vector operations🔗

avoid-insert
  : ∀ {n} {ℓ} {A : Type ℓ}
  → (ρ : Fin n → A)
  → (i : Fin (suc n)) (a : A)
  → (j : Fin (suc n))
  → (i≠j : i ≠ j)
  → (ρ [ i ≔ a ]) j ≡ ρ (avoid i j i≠j)
avoid-insert ρ i a j i≠j with fin-view i | fin-view j
... | zero | zero   = absurd (i≠j refl)
... | zero | suc j  = refl
avoid-insert {suc n} ρ _ a _ _   | suc i | zero = refl
avoid-insert {suc n} ρ _ a _ i≠j | suc i | suc j =
  avoid-insert (ρ ∘ fsuc) i a j (i≠j ∘ ap fsuc)

insert-lookup
  : ∀ {n} {ℓ} {A : Type ℓ}
  → (ρ : Fin n → A)
  → (i : Fin (suc n)) (a : A)
  → (ρ [ i ≔ a ]) i ≡ a
insert-lookup {n = n} ρ i a with fin-view i
... | zero = refl
insert-lookup {n = suc n} ρ _ a | suc i = insert-lookup (ρ ∘ fsuc) i a

delete-insert
  : ∀ {n} {ℓ} {A : Type ℓ}
  → (ρ : Fin n → A)
  → (i : Fin (suc n)) (a : A)
  → ∀ j → delete (ρ [ i ≔ a ]) i j ≡ ρ j
delete-insert ρ i a j with fin-view i | fin-view j
... | zero  | j       = refl
... | suc i | zero    = refl
... | suc i | (suc j) = delete-insert (ρ ∘ fsuc) i a j

insert-delete
  : ∀ {n} {ℓ} {A : Type ℓ}
  → (ρ : Fin (suc n) → A)
  → (i : Fin (suc n)) (a : A)
  → ρ i ≡ a
  → ∀ j → ((delete ρ i) [ i ≔ a ]) j ≡ ρ j
insert-delete ρ i a p j with fin-view i | fin-view j
... | zero  | zero  = sym p
... | zero  | suc j = refl
insert-delete {suc n} ρ _ a p _ | suc i | zero  = refl
insert-delete {suc n} ρ _ a p _ | suc i | suc j = insert-delete (ρ ∘ fsuc) i a p j

ℕ< : Nat → Type
ℕ< n = Σ[ k ∈ Nat ] k Nat.< n

from-ℕ< : ∀ {n} → ℕ< n → Fin n
from-ℕ< (i , p) = fin i ⦃ forget p ⦄

to-ℕ< : ∀ {n} → Fin n → ℕ< n
to-ℕ< (fin i ⦃ forget p ⦄) = i , recover p

fsuc-is-embedding : ∀ {n} → is-embedding (fsuc {n})
fsuc-is-embedding = injective→is-embedding! fsuc-inj