```------------------------------------------------------------------------
-- The Agda standard library
--
-- Equivalence (coinhabitance)
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

module Function.Equivalence where

open import Function using (flip)
open import Function.Equality as F
using (_⟶_; _⟨\$⟩_) renaming (_∘_ to _⟪∘⟫_)
open import Level
open import Relation.Binary
import Relation.Binary.PropositionalEquality as P

------------------------------------------------------------------------
-- Setoid equivalence

record Equivalence {f₁ f₂ t₁ t₂}
(From : Setoid f₁ f₂) (To : Setoid t₁ t₂) :
Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where
field
to   : From ⟶ To
from : To ⟶ From

------------------------------------------------------------------------
-- The set of all equivalences between two sets (i.e. equivalences
-- with propositional equality)

infix 3 _⇔_

_⇔_ : ∀ {f t} → Set f → Set t → Set _
From ⇔ To = Equivalence (P.setoid From) (P.setoid To)

equivalence : ∀ {f t} {From : Set f} {To : Set t} →
(From → To) → (To → From) → From ⇔ To
equivalence to from = record
{ to   = P.→-to-⟶ to
; from = P.→-to-⟶ from
}

------------------------------------------------------------------------
-- Equivalence is an equivalence relation

-- Identity and composition (reflexivity and transitivity).

id : ∀ {s₁ s₂} → Reflexive (Equivalence {s₁} {s₂})
id {x = S} = record
{ to   = F.id
; from = F.id
}

infixr 9 _∘_

_∘_ : ∀ {f₁ f₂ m₁ m₂ t₁ t₂} →
TransFlip (Equivalence {f₁} {f₂} {m₁} {m₂})
(Equivalence {m₁} {m₂} {t₁} {t₂})
(Equivalence {f₁} {f₂} {t₁} {t₂})
f ∘ g = record
{ to   = to   f ⟪∘⟫ to   g
; from = from g ⟪∘⟫ from f
} where open Equivalence

-- Symmetry.

sym : ∀ {f₁ f₂ t₁ t₂} →
Sym (Equivalence {f₁} {f₂} {t₁} {t₂})
(Equivalence {t₁} {t₂} {f₁} {f₂})
sym eq = record
{ from       = to
; to         = from
} where open Equivalence eq

-- For fixed universe levels we can construct setoids.

setoid : (s₁ s₂ : Level) → Setoid (suc (s₁ ⊔ s₂)) (s₁ ⊔ s₂)
setoid s₁ s₂ = record
{ Carrier       = Setoid s₁ s₂
; _≈_           = Equivalence
; isEquivalence = record
{ refl  = id
; sym   = sym
; trans = flip _∘_
}
}

⇔-setoid : (ℓ : Level) → Setoid (suc ℓ) ℓ
⇔-setoid ℓ = record
{ Carrier       = Set ℓ
; _≈_           = _⇔_
; isEquivalence = record
{ refl  = id
; sym   = sym
; trans = flip _∘_
}
}

------------------------------------------------------------------------
-- Transformations

map : ∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂}
{f₁′ f₂′ t₁′ t₂′}
{From′ : Setoid f₁′ f₂′} {To′ : Setoid t₁′ t₂′} →
((From ⟶ To) → (From′ ⟶ To′)) →
((To ⟶ From) → (To′ ⟶ From′)) →
Equivalence From To → Equivalence From′ To′
map t f eq = record { to = t to; from = f from }
where open Equivalence eq

zip : ∀ {f₁₁ f₂₁ t₁₁ t₂₁}
{From₁ : Setoid f₁₁ f₂₁} {To₁ : Setoid t₁₁ t₂₁}
{f₁₂ f₂₂ t₁₂ t₂₂}
{From₂ : Setoid f₁₂ f₂₂} {To₂ : Setoid t₁₂ t₂₂}
{f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →
((From₁ ⟶ To₁) → (From₂ ⟶ To₂) → (From ⟶ To)) →
((To₁ ⟶ From₁) → (To₂ ⟶ From₂) → (To ⟶ From)) →
Equivalence From₁ To₁ → Equivalence From₂ To₂ →
Equivalence From To
zip t f eq₁ eq₂ =
record { to = t (to eq₁) (to eq₂); from = f (from eq₁) (from eq₂) }
where open Equivalence
```