```------------------------------------------------------------------------
-- The Agda standard library
--
-- The reflexive, symmetric and transitive closure of a binary
-- relation (aka the equivalence closure).
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Relation.Binary.Construct.Closure.Equivalence where

open import Function.Base using (flip; id; _∘_; _on_)
open import Level using (Level; _⊔_)
open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)
open import Relation.Binary.Bundles using (Setoid)
open import Relation.Binary.Structures using (IsEquivalence)
open import Relation.Binary.Definitions
using (Reflexive; Transitive; Symmetric)
open import Relation.Binary.Construct.Closure.ReflexiveTransitive as Star
using (Star; ε; _◅◅_; reverse)
open import Relation.Binary.Construct.Closure.Symmetric as SC
using (SymClosure)
import Relation.Binary.Construct.On as On

private
variable
a ℓ ℓ₁ ℓ₂ : Level
A B : Set a
R S : Rel A ℓ

------------------------------------------------------------------------
-- Definition

EqClosure : {A : Set a} → Rel A ℓ → Rel A (a ⊔ ℓ)
EqClosure R = Star (SymClosure R)

------------------------------------------------------------------------
-- Properties

module _ (_∼_ : Rel A ℓ) where

reflexive : Reflexive (EqClosure _∼_)
reflexive = ε

transitive : Transitive (EqClosure _∼_)
transitive = _◅◅_

symmetric : Symmetric (EqClosure _∼_)
symmetric = reverse (SC.symmetric _∼_)

isEquivalence : IsEquivalence (EqClosure _∼_)
isEquivalence = record
{ refl  = reflexive
; sym   = symmetric
; trans = transitive
}

setoid : {A : Set a} (_∼_ : Rel A ℓ) → Setoid a (a ⊔ ℓ)
setoid _∼_ = record
{ _≈_           = EqClosure _∼_
; isEquivalence = isEquivalence _∼_
}

------------------------------------------------------------------------
-- Operations

-- A generalised variant of `map` which allows the index type to change.
gmap : (f : A → B) → R =[ f ]⇒ S → EqClosure R =[ f ]⇒ EqClosure S
gmap f = Star.gmap f ∘ SC.gmap f

map : R ⇒ S → EqClosure R ⇒ EqClosure S
map = gmap id

fold : IsEquivalence S → R ⇒ S → EqClosure R ⇒ S
fold S-equiv R⇒S = Star.fold _ (trans ∘ SC.fold sym R⇒S) refl
where open IsEquivalence S-equiv

-- A generalised variant of `fold`.
gfold : IsEquivalence S → (f : A → B) → R =[ f ]⇒ S → EqClosure R =[ f ]⇒ S
gfold S-equiv f R⇒S = fold (On.isEquivalence f S-equiv) R⇒S

-- `return` could also be called `singleton`.
return : R ⇒ EqClosure R
return = Star.return ∘ SC.return

-- `join` could also be called `concat`.
join : EqClosure (EqClosure R) ⇒ EqClosure R
join = fold (isEquivalence _) id

infix 10 _⋆

_⋆ : R ⇒ EqClosure S → EqClosure R ⇒ EqClosure S
_⋆ f m = join (map f m)
```