```------------------------------------------------------------------------
-- The Agda standard library
--
-- Surjections
------------------------------------------------------------------------

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

-- Note: use of the standard function hierarchy is encouraged. The
-- module `Function` re-exports `Surjective`, `IsSurjection` and
-- `Surjection`. The alternative definitions found in this file will
-- eventually be deprecated.

module Function.Surjection where

open import Level
open import Function.Equality as F
using (_⟶_) renaming (_∘_ to _⟪∘⟫_)
open import Function.Equivalence using (Equivalence)
open import Function.Injection           hiding (id; _∘_; injection)
open import Function.LeftInverse as Left hiding (id; _∘_)
open import Data.Product
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as P using (_≡_)

------------------------------------------------------------------------
-- Surjective functions.

record Surjective {f₁ f₂ t₁ t₂}
{From : Setoid f₁ f₂} {To : Setoid t₁ t₂}
(to : From ⟶ To) :
Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where
field
from             : To ⟶ From
right-inverse-of : from RightInverseOf to

------------------------------------------------------------------------
-- The set of all surjections from one setoid to another.

record Surjection {f₁ f₂ t₁ t₂}
(From : Setoid f₁ f₂) (To : Setoid t₁ t₂) :
Set (f₁ ⊔ f₂ ⊔ t₁ ⊔ t₂) where
field
to         : From ⟶ To
surjective : Surjective to

open Surjective surjective public

right-inverse : RightInverse From To
right-inverse = record
{ to              = from
; from            = to
; left-inverse-of = right-inverse-of
}

open LeftInverse right-inverse public
using () renaming (to-from to from-to)

injective : Injective from
injective = LeftInverse.injective right-inverse

injection : Injection To From
injection = LeftInverse.injection right-inverse

equivalence : Equivalence From To
equivalence = record
{ to   = to
; from = from
}

-- Right inverses can be turned into surjections.

fromRightInverse :
∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →
RightInverse From To → Surjection From To
fromRightInverse r = record
{ to         = from
; surjective = record
{ from             = to
; right-inverse-of = left-inverse-of
}
} where open LeftInverse r

------------------------------------------------------------------------
-- The set of all surjections from one set to another (i.e. sujections
-- with propositional equality)

infix 3 _↠_

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

surjection : ∀ {f t} {From : Set f} {To : Set t} →
(to : From → To) (from : To → From) →
(∀ x → to (from x) ≡ x) →
From ↠ To
surjection to from surjective = record
{ to         = P.→-to-⟶ to
; surjective = record
{ from             = P.→-to-⟶ from
; right-inverse-of = surjective
}
}

------------------------------------------------------------------------
-- Identity and composition.

id : ∀ {s₁ s₂} {S : Setoid s₁ s₂} → Surjection S S
id {S = S} = record
{ to         = F.id
; surjective = record
{ from             = LeftInverse.to              id′
; right-inverse-of = LeftInverse.left-inverse-of id′
}
} where id′ = Left.id {S = S}

infixr 9 _∘_

_∘_ : ∀ {f₁ f₂ m₁ m₂ t₁ t₂}
{F : Setoid f₁ f₂} {M : Setoid m₁ m₂} {T : Setoid t₁ t₂} →
Surjection M T → Surjection F M → Surjection F T
f ∘ g = record
{ to         = to f ⟪∘⟫ to g
; surjective = record
{ from             = LeftInverse.to              g∘f
; right-inverse-of = LeftInverse.left-inverse-of g∘f
}
}
where
open Surjection
g∘f = Left._∘_ (right-inverse g) (right-inverse f)
```