```------------------------------------------------------------------------
-- Pointwise lifting of binary relations to sigma types
------------------------------------------------------------------------

{-# OPTIONS --universe-polymorphism #-}

module Relation.Binary.Sigma.Pointwise where

open import Data.Product as Prod
open import Level
open import Function
import Function.Equality as F
open import Function.Equivalence as Eq
using (Equivalent; _⇔_; module Equivalent)
renaming (_∘_ to _⟨∘⟩_)
open import Function.Inverse as Inv
using (Inverse; _⇿_; module Inverse; Isomorphism)
renaming (_∘_ to _⟪∘⟫_)
open import Function.LeftInverse
using (_LeftInverseOf_; _RightInverseOf_)
import Relation.Binary as B
open import Relation.Binary.Indexed as I using (_at_)
import Relation.Binary.HeterogeneousEquality as H
import Relation.Binary.PropositionalEquality as P

------------------------------------------------------------------------
-- Pointwise lifting

infixr 4 _,_

data REL {a₁ a₂ b₁ b₂ ℓ₁ ℓ₂}
{A₁ : Set a₁} (B₁ : A₁ → Set b₁)
{A₂ : Set a₂} (B₂ : A₂ → Set b₂)
(_R₁_ : B.REL A₁ A₂ ℓ₁) (_R₂_ : I.REL B₁ B₂ ℓ₂) :
B.REL (Σ A₁ B₁) (Σ A₂ B₂) (a₁ ⊔ a₂ ⊔ b₁ ⊔ b₂ ⊔ ℓ₁ ⊔ ℓ₂) where
_,_ : {x₁ : A₁} {y₁ : B₁ x₁} {x₂ : A₂} {y₂ : B₂ x₂}
(x₁Rx₂ : x₁ R₁ x₂) (y₁Ry₂ : y₁ R₂ y₂) →
REL B₁ B₂ _R₁_ _R₂_ (x₁ , y₁) (x₂ , y₂)

Rel : ∀ {a b ℓ₁ ℓ₂} {A : Set a} (B : A → Set b)
(_R₁_ : B.Rel A ℓ₁) (_R₂_ : I.Rel B ℓ₂) → B.Rel (Σ A B) _
Rel B = REL B B

------------------------------------------------------------------------
-- Rel preserves many properties

private
module Dummy {a b ℓ₁ ℓ₂} {A : Set a} {B : A → Set b}
{R₁ : B.Rel A ℓ₁} {R₂ : I.Rel B ℓ₂} where

refl : B.Reflexive R₁ → I.Reflexive B R₂ →
B.Reflexive (Rel B R₁ R₂)
refl refl₁ refl₂ {x = (x , y)} = (refl₁ , refl₂)

symmetric : B.Symmetric R₁ → I.Symmetric B R₂ →
B.Symmetric (Rel B R₁ R₂)
symmetric sym₁ sym₂ (x₁Rx₂ , y₁Ry₂) = (sym₁ x₁Rx₂ , sym₂ y₁Ry₂)

transitive : B.Transitive R₁ → I.Transitive B R₂ →
B.Transitive (Rel B R₁ R₂)
transitive trans₁ trans₂ (x₁Rx₂ , y₁Ry₂) (x₂Rx₃ , y₂Ry₃) =
(trans₁ x₁Rx₂ x₂Rx₃ , trans₂ y₁Ry₂ y₂Ry₃)

isEquivalence : B.IsEquivalence R₁ → I.IsEquivalence B R₂ →
B.IsEquivalence (Rel B R₁ R₂)
isEquivalence eq₁ eq₂ = record
{ refl  = refl (B.IsEquivalence.refl eq₁)
(I.IsEquivalence.refl eq₂)
; sym   = symmetric (B.IsEquivalence.sym eq₁)
(I.IsEquivalence.sym eq₂)
; trans = transitive (B.IsEquivalence.trans eq₁)
(I.IsEquivalence.trans eq₂)
}

open Dummy public

setoid : ∀ {b₁ b₂ i₁ i₂} →
(A : B.Setoid b₁ b₂) → I.Setoid (B.Setoid.Carrier A) i₁ i₂ →
B.Setoid _ _
setoid s₁ s₂ = record
{ isEquivalence = isEquivalence (B.Setoid.isEquivalence s₁)
(I.Setoid.isEquivalence s₂)
}

------------------------------------------------------------------------
-- The propositional equality setoid over sigma types can be
-- decomposed using Rel

Rel⇿≡ : ∀ {a b} {A : Set a} {B : A → Set b} →
Inverse (setoid (P.setoid A) (H.indexedSetoid B))
(P.setoid (Σ A B))
Rel⇿≡ {a} {b} {A} {B} = record
{ to         = record { _⟨\$⟩_ = id; cong = to-cong   }
; from       = record { _⟨\$⟩_ = id; cong = from-cong }
; inverse-of = record
{ left-inverse-of  = uncurry (λ _ _ → (P.refl , H.refl))
; right-inverse-of = λ _ → P.refl
}
}
where
open I using (_=[_]⇒_)

to-cong : Rel B P._≡_ (λ x y → H._≅_ x y) =[ id {a = a ⊔ b} ]⇒ P._≡_
to-cong (P.refl , H.refl) = P.refl

from-cong : P._≡_ =[ id {a = a ⊔ b} ]⇒ Rel B P._≡_ (λ x y → H._≅_ x y)
from-cong {i = (x , y)} P.refl = (P.refl , H.refl)

------------------------------------------------------------------------
-- Equivalences and inverses are also preserved

equivalent :
∀ {i} {I : Set i}
{f₁ f₂ t₁ t₂} {From : I.Setoid I f₁ f₂} {To : I.Setoid I t₁ t₂} →
(∀ {i} → Equivalent (From at i) (To at i)) →
Equivalent (setoid (P.setoid I) From) (setoid (P.setoid I) To)
equivalent {I = I} {From = F} {T} F⇔T = record
{ to   = record { _⟨\$⟩_ = to;   cong = to-cong   }
; from = record { _⟨\$⟩_ = from; cong = from-cong }
}
where
open B.Setoid (setoid (P.setoid I) F) using () renaming (_≈_ to _≈F_)
open B.Setoid (setoid (P.setoid I) T) using () renaming (_≈_ to _≈T_)
open B using (_=[_]⇒_)

to = Prod.map id (F._⟨\$⟩_ (Equivalent.to F⇔T))

to-cong : _≈F_ =[ to ]⇒ _≈T_
to-cong (P.refl , ∼) = (P.refl , F.cong (Equivalent.to F⇔T) ∼)

from = Prod.map id (F._⟨\$⟩_ (Equivalent.from F⇔T))

from-cong : _≈T_ =[ from ]⇒ _≈F_
from-cong (P.refl , ∼) = (P.refl , F.cong (Equivalent.from F⇔T) ∼)

⇔ : ∀ {a b₁ b₂} {A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ {x} → B₁ x ⇔ B₂ x) → Σ A B₁ ⇔ Σ A B₂
⇔ {B₁ = B₁} {B₂} B₁⇔B₂ =
Inverse.equivalent (Rel⇿≡ {B = B₂}) ⟨∘⟩
equivalent (λ {x} →
Inverse.equivalent (H.≡⇿≅ B₂) ⟨∘⟩
B₁⇔B₂ {x} ⟨∘⟩
Inverse.equivalent (Inv.sym (H.≡⇿≅ B₁))) ⟨∘⟩
Eq.sym (Inverse.equivalent (Rel⇿≡ {B = B₁}))

inverse :
∀ {i} {I : Set i}
{f₁ f₂ t₁ t₂} {From : I.Setoid I f₁ f₂} {To : I.Setoid I t₁ t₂} →
(∀ {i} → Inverse (From at i) (To at i)) →
Inverse (setoid (P.setoid I) From) (setoid (P.setoid I) To)
inverse {I = I} {From = F} {T} F⇿T = record
{ to         = Equivalent.to   eq
; from       = Equivalent.from eq
; inverse-of = record
{ left-inverse-of  = left
; right-inverse-of = right
}
}
where
eq = equivalent (Inverse.equivalent F⇿T)

left : Equivalent.from eq LeftInverseOf Equivalent.to eq
left (x , y) = (P.refl , Inverse.left-inverse-of F⇿T y)

right : Equivalent.from eq RightInverseOf Equivalent.to eq
right (x , y) = (P.refl , Inverse.right-inverse-of F⇿T y)

⇿ : ∀ {a b₁ b₂} {A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ {x} → B₁ x ⇿ B₂ x) → Σ A B₁ ⇿ Σ A B₂
⇿ {B₁ = B₁} {B₂} B₁⇿B₂ =
Rel⇿≡ {B = B₂} ⟪∘⟫
inverse (λ {x} → H.≡⇿≅ B₂ ⟪∘⟫ B₁⇿B₂ {x} ⟪∘⟫ Inv.sym (H.≡⇿≅ B₁)) ⟪∘⟫
Inv.sym (Rel⇿≡ {B = B₁})

cong : ∀ {k a b₁ b₂} {A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ {x} → Isomorphism k (B₁ x) (B₂ x)) →
Isomorphism k (Σ A B₁) (Σ A B₂)
cong {k = Inv.equivalent} = ⇔
cong {k = Inv.inverse}    = ⇿
```