Structures.agda

------------------------------------------------------------------------
-- The Agda standard library
--
-- Structures for types of functions
------------------------------------------------------------------------

-- The contents of this file should usually be accessed from `Function`.

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

open import Relation.Binary.Core using (Rel)
open import Relation.Binary.Bundles using (Setoid)
open import Relation.Binary.Structures using (IsEquivalence)

module Function.Structures {a b ℓ₁ ℓ₂}
  {A : Set a} (_≈₁_ : Rel A ℓ₁) -- Equality over the domain
  {B : Set b} (_≈₂_ : Rel B ℓ₂) -- Equality over the codomain
  where

open import Data.Product.Base as Product using (∃; _×_; _,_)
open import Function.Base
open import Function.Definitions
open import Function.Definitions.Strict
open import Level using (_⊔_)

------------------------------------------------------------------------
-- One element structures
------------------------------------------------------------------------

record IsCongruent (to : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
  field
    cong           : Congruent _≈₁_ _≈₂_ to
    isEquivalence₁ : IsEquivalence _≈₁_
    isEquivalence₂ : IsEquivalence _≈₂_

  module Eq₁ where

    setoid : Setoid a ℓ₁
    setoid = record
      { isEquivalence = isEquivalence₁
      }

    open Setoid setoid public

  module Eq₂ where

    setoid : Setoid b ℓ₂
    setoid = record
      { isEquivalence = isEquivalence₂
      }

    open Setoid setoid public


record IsInjection (to : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
  field
    isCongruent : IsCongruent to
    injective   : Injective _≈₁_ _≈₂_ to

  open IsCongruent isCongruent public


record IsSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
  field
    isCongruent : IsCongruent f
    surjective  : Surjective _≈₁_ _≈₂_ f

  open IsCongruent isCongruent public

  strictlySurjective : StrictlySurjective _≈₂_ f
  strictlySurjective x = Product.map₂ (λ v → v Eq₁.refl) (surjective x)


record IsBijection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
  field
    isInjection : IsInjection f
    surjective  : Surjective _≈₁_ _≈₂_ f

  open IsInjection isInjection public

  bijective : Bijective _≈₁_ _≈₂_ f
  bijective = injective , surjective

  isSurjection : IsSurjection f
  isSurjection = record
    { isCongruent = isCongruent
    ; surjective  = surjective
    }

  open IsSurjection isSurjection public
    using (strictlySurjective)


------------------------------------------------------------------------
-- Two element structures
------------------------------------------------------------------------

record IsLeftInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
  field
    isCongruent  : IsCongruent to
    from-cong    : Congruent _≈₂_ _≈₁_ from
    inverseˡ     : Inverseˡ _≈₁_ _≈₂_ to from

  open IsCongruent isCongruent public
    renaming (cong to to-cong)

  strictlyInverseˡ : StrictlyInverseˡ _≈₂_ to from
  strictlyInverseˡ x = inverseˡ Eq₁.refl

  isSurjection : IsSurjection to
  isSurjection = record
    { isCongruent = isCongruent
    ; surjective = λ y → from y , inverseˡ
    }


record IsRightInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
  field
    isCongruent : IsCongruent to
    from-cong   : Congruent _≈₂_ _≈₁_ from
    inverseʳ    : Inverseʳ _≈₁_ _≈₂_ to from

  open IsCongruent isCongruent public
    renaming (cong to to-cong)

  strictlyInverseʳ : StrictlyInverseʳ _≈₁_ to from
  strictlyInverseʳ x = inverseʳ Eq₂.refl


record IsInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
  field
    isLeftInverse : IsLeftInverse to from
    inverseʳ      : Inverseʳ _≈₁_ _≈₂_ to from

  open IsLeftInverse isLeftInverse public

  isRightInverse : IsRightInverse to from
  isRightInverse = record
    { isCongruent = isCongruent
    ; from-cong   = from-cong
    ; inverseʳ    = inverseʳ
    }

  open IsRightInverse isRightInverse public
    using (strictlyInverseʳ)

  inverse : Inverseᵇ _≈₁_ _≈₂_ to from
  inverse = inverseˡ , inverseʳ


------------------------------------------------------------------------
-- Three element structures
------------------------------------------------------------------------

record IsBiEquivalence
  (to : A → B) (from₁ : B → A) (from₂ : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
  field
    to-isCongruent : IsCongruent to
    from₁-cong    : Congruent _≈₂_ _≈₁_ from₁
    from₂-cong    : Congruent _≈₂_ _≈₁_ from₂

  open IsCongruent to-isCongruent public
    renaming (cong to to-cong₁)


record IsBiInverse
  (to : A → B) (from₁ : B → A) (from₂ : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
  field
    to-isCongruent : IsCongruent to
    from₁-cong     : Congruent _≈₂_ _≈₁_ from₁
    from₂-cong     : Congruent _≈₂_ _≈₁_ from₂
    inverseˡ       : Inverseˡ _≈₁_ _≈₂_ to from₁
    inverseʳ       : Inverseʳ _≈₁_ _≈₂_ to from₂

  open IsCongruent to-isCongruent public
    renaming (cong to to-cong)


------------------------------------------------------------------------
-- Other
------------------------------------------------------------------------

-- See the comment on `SplitSurjection` in `Function.Bundles` for an
-- explanation of (split) surjections.
record IsSplitSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
  field
    from : B → A
    isLeftInverse : IsLeftInverse f from

  open IsLeftInverse isLeftInverse public