{-# OPTIONS --cubical-compatible --safe #-}
module Function.Construct.Composition where
open import Data.Product.Base as Product using (_,_)
open import Function.Base using (_∘_; flip)
open import Function.Bundles
open import Function.Definitions
open import Function.Structures
open import Level using (Level)
open import Relation.Binary.Core using (Rel)
open import Relation.Binary.Bundles using (Setoid)
open import Relation.Binary.Definitions using (Transitive)
private
variable
a b c ℓ₁ ℓ₂ ℓ₃ : Level
A B C : Set a
module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) (≈₃ : Rel C ℓ₃)
{f : A → B} {g : B → C}
where
congruent : Congruent ≈₁ ≈₂ f → Congruent ≈₂ ≈₃ g →
Congruent ≈₁ ≈₃ (g ∘ f)
congruent f-cong g-cong = g-cong ∘ f-cong
injective : Injective ≈₁ ≈₂ f → Injective ≈₂ ≈₃ g →
Injective ≈₁ ≈₃ (g ∘ f)
injective f-inj g-inj = f-inj ∘ g-inj
surjective : Surjective ≈₁ ≈₂ f → Surjective ≈₂ ≈₃ g →
Surjective ≈₁ ≈₃ (g ∘ f)
surjective f-sur g-sur x with g-sur x
... | y , gproof with f-sur y
... | z , fproof = z , gproof ∘ fproof
bijective : Bijective ≈₁ ≈₂ f → Bijective ≈₂ ≈₃ g →
Bijective ≈₁ ≈₃ (g ∘ f)
bijective = Product.zip′ injective surjective
module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) (≈₃ : Rel C ℓ₃)
{f : A → B} {f⁻¹ : B → A} {g : B → C} {g⁻¹ : C → B}
where
inverseˡ : Inverseˡ ≈₁ ≈₂ f f⁻¹ → Inverseˡ ≈₂ ≈₃ g g⁻¹ →
Inverseˡ ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)
inverseˡ f-inv g-inv = g-inv ∘ f-inv
inverseʳ : Inverseʳ ≈₁ ≈₂ f f⁻¹ → Inverseʳ ≈₂ ≈₃ g g⁻¹ →
Inverseʳ ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)
inverseʳ f-inv g-inv = f-inv ∘ g-inv
inverseᵇ : Inverseᵇ ≈₁ ≈₂ f f⁻¹ → Inverseᵇ ≈₂ ≈₃ g g⁻¹ →
Inverseᵇ ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)
inverseᵇ = Product.zip′ inverseˡ inverseʳ
module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {≈₃ : Rel C ℓ₃}
{f : A → B} {g : B → C}
where
isCongruent : IsCongruent ≈₁ ≈₂ f → IsCongruent ≈₂ ≈₃ g →
IsCongruent ≈₁ ≈₃ (g ∘ f)
isCongruent f-cong g-cong = record
{ cong = G.cong ∘ F.cong
; isEquivalence₁ = F.isEquivalence₁
; isEquivalence₂ = G.isEquivalence₂
} where module F = IsCongruent f-cong; module G = IsCongruent g-cong
isInjection : IsInjection ≈₁ ≈₂ f → IsInjection ≈₂ ≈₃ g →
IsInjection ≈₁ ≈₃ (g ∘ f)
isInjection f-inj g-inj = record
{ isCongruent = isCongruent F.isCongruent G.isCongruent
; injective = injective ≈₁ ≈₂ ≈₃ F.injective G.injective
} where module F = IsInjection f-inj; module G = IsInjection g-inj
isSurjection : IsSurjection ≈₁ ≈₂ f → IsSurjection ≈₂ ≈₃ g →
IsSurjection ≈₁ ≈₃ (g ∘ f)
isSurjection f-surj g-surj = record
{ isCongruent = isCongruent F.isCongruent G.isCongruent
; surjective = surjective ≈₁ ≈₂ ≈₃ F.surjective G.surjective
} where module F = IsSurjection f-surj; module G = IsSurjection g-surj
isBijection : IsBijection ≈₁ ≈₂ f → IsBijection ≈₂ ≈₃ g →
IsBijection ≈₁ ≈₃ (g ∘ f)
isBijection f-bij g-bij = record
{ isInjection = isInjection F.isInjection G.isInjection
; surjective = surjective ≈₁ ≈₂ ≈₃ F.surjective G.surjective
} where module F = IsBijection f-bij; module G = IsBijection g-bij
module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {≈₃ : Rel C ℓ₃}
{f : A → B} {g : B → C} {f⁻¹ : B → A} {g⁻¹ : C → B}
where
isLeftInverse : IsLeftInverse ≈₁ ≈₂ f f⁻¹ → IsLeftInverse ≈₂ ≈₃ g g⁻¹ →
IsLeftInverse ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)
isLeftInverse f-invˡ g-invˡ = record
{ isCongruent = isCongruent F.isCongruent G.isCongruent
; from-cong = congruent ≈₃ ≈₂ ≈₁ G.from-cong F.from-cong
; inverseˡ = inverseˡ ≈₁ ≈₂ ≈₃ F.inverseˡ G.inverseˡ
} where module F = IsLeftInverse f-invˡ; module G = IsLeftInverse g-invˡ
isRightInverse : IsRightInverse ≈₁ ≈₂ f f⁻¹ → IsRightInverse ≈₂ ≈₃ g g⁻¹ →
IsRightInverse ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)
isRightInverse f-invʳ g-invʳ = record
{ isCongruent = isCongruent F.isCongruent G.isCongruent
; from-cong = congruent ≈₃ ≈₂ ≈₁ G.from-cong F.from-cong
; inverseʳ = inverseʳ ≈₁ ≈₂ ≈₃ F.inverseʳ G.inverseʳ
} where module F = IsRightInverse f-invʳ; module G = IsRightInverse g-invʳ
isInverse : IsInverse ≈₁ ≈₂ f f⁻¹ → IsInverse ≈₂ ≈₃ g g⁻¹ →
IsInverse ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)
isInverse f-inv g-inv = record
{ isLeftInverse = isLeftInverse F.isLeftInverse G.isLeftInverse
; inverseʳ = inverseʳ ≈₁ ≈₂ ≈₃ F.inverseʳ G.inverseʳ
} where module F = IsInverse f-inv; module G = IsInverse g-inv
module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} {T : Setoid c ℓ₃} where
open Setoid renaming (_≈_ to ≈)
function : Func R S → Func S T → Func R T
function f g = record
{ to = G.to ∘ F.to
; cong = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong
} where module F = Func f; module G = Func g
injection : Injection R S → Injection S T → Injection R T
injection inj₁ inj₂ = record
{ to = G.to ∘ F.to
; cong = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong
; injective = injective (≈ R) (≈ S) (≈ T) F.injective G.injective
} where module F = Injection inj₁; module G = Injection inj₂
surjection : Surjection R S → Surjection S T → Surjection R T
surjection surj₁ surj₂ = record
{ to = G.to ∘ F.to
; cong = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong
; surjective = surjective (≈ R) (≈ S) (≈ T) F.surjective G.surjective
} where module F = Surjection surj₁; module G = Surjection surj₂
bijection : Bijection R S → Bijection S T → Bijection R T
bijection bij₁ bij₂ = record
{ to = G.to ∘ F.to
; cong = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong
; bijective = bijective (≈ R) (≈ S) (≈ T) F.bijective G.bijective
} where module F = Bijection bij₁; module G = Bijection bij₂
equivalence : Equivalence R S → Equivalence S T → Equivalence R T
equivalence equiv₁ equiv₂ = record
{ to = G.to ∘ F.to
; from = F.from ∘ G.from
; to-cong = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong
; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong
} where module F = Equivalence equiv₁; module G = Equivalence equiv₂
leftInverse : LeftInverse R S → LeftInverse S T → LeftInverse R T
leftInverse invˡ₁ invˡ₂ = record
{ to = G.to ∘ F.to
; from = F.from ∘ G.from
; to-cong = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong
; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong
; inverseˡ = inverseˡ (≈ R) (≈ S) (≈ T) F.inverseˡ G.inverseˡ
} where module F = LeftInverse invˡ₁; module G = LeftInverse invˡ₂
rightInverse : RightInverse R S → RightInverse S T → RightInverse R T
rightInverse invʳ₁ invʳ₂ = record
{ to = G.to ∘ F.to
; from = F.from ∘ G.from
; to-cong = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong
; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong
; inverseʳ = inverseʳ (≈ R) (≈ S) (≈ T) F.inverseʳ G.inverseʳ
} where module F = RightInverse invʳ₁; module G = RightInverse invʳ₂
inverse : Inverse R S → Inverse S T → Inverse R T
inverse inv₁ inv₂ = record
{ to = G.to ∘ F.to
; from = F.from ∘ G.from
; to-cong = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong
; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong
; inverse = inverseᵇ (≈ R) (≈ S) (≈ T) F.inverse G.inverse
} where module F = Inverse inv₁; module G = Inverse inv₂
infix 8 _⟶-∘_ _↣-∘_ _↠-∘_ _⤖-∘_ _⇔-∘_ _↩-∘_ _↪-∘_ _↔-∘_
_⟶-∘_ : (B ⟶ C) → (A ⟶ B) → (A ⟶ C)
_⟶-∘_ = flip function
_↣-∘_ : B ↣ C → A ↣ B → A ↣ C
_↣-∘_ = flip injection
_↠-∘_ : B ↠ C → A ↠ B → A ↠ C
_↠-∘_ = flip surjection
_⤖-∘_ : B ⤖ C → A ⤖ B → A ⤖ C
_⤖-∘_ = flip bijection
_⇔-∘_ : B ⇔ C → A ⇔ B → A ⇔ C
_⇔-∘_ = flip equivalence
_↩-∘_ : B ↩ C → A ↩ B → A ↩ C
_↩-∘_ = flip leftInverse
_↪-∘_ : B ↪ C → A ↪ B → A ↪ C
_↪-∘_ = flip rightInverse
_↔-∘_ : B ↔ C → A ↔ B → A ↔ C
_↔-∘_ = flip inverse
infix 8 _∘-⟶_ _∘-↣_ _∘-↠_ _∘-⤖_ _∘-⇔_ _∘-↩_ _∘-↪_ _∘-↔_
_∘-⟶_ = _⟶-∘_
{-# WARNING_ON_USAGE _∘-⟶_
"Warning: _∘-⟶_ was deprecated in v2.0.
Please use _⟶-∘_ instead."
#-}
_∘-↣_ = _↣-∘_
{-# WARNING_ON_USAGE _∘-↣_
"Warning: _∘-↣_ was deprecated in v2.0.
Please use _↣-∘_ instead."
#-}
_∘-↠_ = _↠-∘_
{-# WARNING_ON_USAGE _∘-↠_
"Warning: _∘-↠_ was deprecated in v2.0.
Please use _↠-∘_ instead."
#-}
_∘-⤖_ = _⤖-∘_
{-# WARNING_ON_USAGE _∘-⤖_
"Warning: _∘-⤖_ was deprecated in v2.0.
Please use _⤖-∘_ instead."
#-}
_∘-⇔_ = _⇔-∘_
{-# WARNING_ON_USAGE _∘-⇔_
"Warning: _∘-⇔_ was deprecated in v2.0.
Please use _⇔-∘_ instead."
#-}
_∘-↩_ = _↩-∘_
{-# WARNING_ON_USAGE _∘-↩_
"Warning: _∘-↩_ was deprecated in v2.0.
Please use _↩-∘_ instead."
#-}
_∘-↪_ = _↪-∘_
{-# WARNING_ON_USAGE _∘-↪_
"Warning: _∘-↪_ was deprecated in v2.0.
Please use _↪-∘_ instead."
#-}
_∘-↔_ = _↔-∘_
{-# WARNING_ON_USAGE _∘-↔_
"Warning: _∘-↔_ was deprecated in v2.0.
Please use _↔-∘_ instead."
#-}