---
layout: default
title : "Setoid.Functions.Injective module"
date : "2021-09-13"
author: "the agda-algebras development team"
---

### Injective functions on setoids

This is the [Setoid.Functions.Injective][] module of the [agda-algebras][] library.

We say that a function `f : A → B` from one setoid (A , ≈₀) to another (B , ≈₁) is *injective* (or *monic*) provided the following implications hold:  ∀ a₀ a₁ if f ⟨$⟩ a₀ ≈₁ f ⟨$⟩ a₁, then a₀ ≈₀ a₁.

<!--
```agda
{-# OPTIONS --cubical-compatible --exact-split --safe #-}

open import Relation.Binary using ( Setoid )

module Setoid.Functions.Injective where

-- Imports from Agda and the Agda Standard Library -------------
open import Agda.Primitive    using ( _⊔_ ; Level )  renaming ( Set to Type )
open import Function.Bundles  using ( Injection )    renaming ( Func to _⟶_ )
open import Function.Base     using ( _∘_ ; id )
open import Relation.Binary   using ( _Preserves_⟶_ )
open import Relation.Binary   using ( Rel )

open import Function.Definitions using (Injective)

-- Imports from agda-algebras -----------------------------------------------
open import Setoid.Functions.Basic     using ( 𝑖𝑑 ) renaming ( _⊙_ to _⟨⊙⟩_ )
open import Setoid.Functions.Inverses  using ( Image_∋_ ; Inv )

private variable a b c α β γ ℓ₁ ℓ₂ ℓ₃ : Level
```
-->

A function `f : A ⟶ B` from one setoid `(A , ≈₀)` to another
`(B , ≈₁)` is called *injective* provided `∀ a₀ a₁`, if `f ⟨$⟩ a₀ ≈₁ f ⟨$⟩
a₁`, then `a₀ ≈₀ a₁`.  The [Agda Standard Library][] defines a type representing
injective functions on bare types and we use this type (called `Injective`) to
define our own type representing the property of being an injective function on
setoids (called `IsInjective`).

```agda
module _ {𝑨 : Setoid a α}{𝑩 : Setoid b β} where
  open Setoid 𝑨 using ()               renaming (Carrier to A; _≈_ to _≈₁_)
  open Setoid 𝑩 using ( trans ; sym )  renaming (Carrier to B; _≈_ to _≈₂_)

  open Injection {From = 𝑨}{To = 𝑩} using ( function ; injective ) renaming (to to _⟨$⟩_)
  open _⟶_ {a = a}{α}{b}{β}{From = 𝑨}{To = 𝑩} renaming (to to _⟨$⟩_ )
  IsInjective : (𝑨 ⟶ 𝑩) → Type (a ⊔ α ⊔ β)
  IsInjective f = Injective _≈₁_ _≈₂_ (_⟨$⟩_ f)

  open Image_∋_

  -- Inverse of an injective function preserves setoid equalities
  LeftInvPreserves≈ :  (F : Injection 𝑨 𝑩) {b₀ b₁ : B}
    (u : Image (function F) ∋ b₀) (v : Image (function F) ∋ b₁)
    → b₀ ≈₂ b₁ → Inv (function F) u ≈₁ Inv (function F) v

  LeftInvPreserves≈ F (eq a₀ x₀) (eq a₁ x₁) bb = Goal
    where
    fa₀≈fa₁ : F ⟨$⟩ a₀ ≈₂ F ⟨$⟩ a₁
    fa₀≈fa₁ = trans (sym x₀) (trans bb x₁)

    Goal : a₀ ≈₁ a₁
    Goal = injective F fa₀≈fa₁
```

Proving that the composition of injective functions is again injective
is simply a matter of composing the two assumed witnesses to injectivity.
(Note that here we are viewing the maps as functions on the underlying carriers
of the setoids; an alternative for setoid functions, called `∘-injective`, is proved below.)

```agda
module _
  {A : Type a}(_≈₁_ : Rel A α)
  {B : Type b}(_≈₂_ : Rel B β)
  {C : Type c}(_≈₃_ : Rel C γ) where

  ∘-injective-bare : {f : A → B} {g : B → C}
    → Injective _≈₁_ _≈₂_ f → Injective _≈₂_ _≈₃_ g
    → Injective _≈₁_ _≈₃_ (g ∘ f)

  ∘-injective-bare finj ginj = finj ∘ ginj

module _ {𝑨 : Setoid a α}{𝑩 : Setoid b β}{𝑪 : Setoid c γ} where

  ⊙-injective :  (f : 𝑨 ⟶ 𝑩)(g : 𝑩 ⟶ 𝑪)
   →             IsInjective f → IsInjective g
   →             IsInjective (g ⟨⊙⟩ f)

  ⊙-injective _ _ finj ginj = finj ∘ ginj

  ⊙-injection : Injection 𝑨 𝑩 → Injection 𝑩 𝑪 → Injection 𝑨 𝑪
  ⊙-injection fi gi = record
   { to = to gi ∘ to fi
   ; cong = cong gi ∘ cong fi
   ; injective = ⊙-injective (function fi) (function gi) (injective fi) (injective gi)
   }
   where open Injection

id-is-injective : {𝑨 : Setoid a α} → IsInjective{𝑨 = 𝑨}{𝑨} 𝑖𝑑
id-is-injective = id
```