{-# OPTIONS --without-K --safe #-}
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 (_≡_)
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
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
}
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
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
}
}
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)