```------------------------------------------------------------------------
-- The Agda standard library
--
-- Pointwise lifting of binary relations to sigma types
------------------------------------------------------------------------

module Data.Product.Relation.SigmaPointwise where

open import Data.Product as Prod
open import Level
open import Function
open import Function.Equality as F using (_⟶_; _⟨\$⟩_)
open import Function.Equivalence as Eq
using (Equivalence; _⇔_; module Equivalence)
open import Function.Injection as Inj
using (Injection; _↣_; module Injection; Injective)
open import Function.Inverse as Inv
using (Inverse; _↔_; module Inverse)
open import Function.LeftInverse as LeftInv
using (LeftInverse; _↞_; module LeftInverse;
_LeftInverseOf_; _RightInverseOf_)
open import Function.Related as Related
using (_∼[_]_; lam; app-←; app-↢)
open import Function.Surjection as Surj
using (Surjection; _↠_; module Surjection)
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

module _ {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)
}

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)

------------------------------------------------------------------------
-- Some properties related to "relatedness"

⟶ : ∀ {a₁ a₂ b₁ b₁′ b₂ b₂′}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : I.Setoid A₁ b₁ b₁′} (B₂ : I.Setoid A₂ b₂ b₂′)
(f : A₁ → A₂) → (∀ {x} → (B₁ at x) ⟶ (B₂ at f x)) →
setoid (P.setoid A₁) B₁ ⟶ setoid (P.setoid A₂) B₂
⟶ {A₁ = A₁} {A₂} {B₁} B₂ f g = record
{ _⟨\$⟩_ = fg
; cong  = fg-cong
}
where
open B.Setoid (setoid (P.setoid A₁) B₁)
using () renaming (_≈_ to _≈₁_)
open B.Setoid (setoid (P.setoid A₂) B₂)
using () renaming (_≈_ to _≈₂_)
open B using (_=[_]⇒_)

fg = Prod.map f (_⟨\$⟩_ g)

fg-cong : _≈₁_ =[ fg ]⇒ _≈₂_
fg-cong (P.refl , ∼) = (P.refl , F.cong g ∼)

equivalence :
∀ {a₁ a₂ b₁ b₁′ b₂ b₂′}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : I.Setoid A₁ b₁ b₁′} {B₂ : I.Setoid A₂ b₂ b₂′}
(A₁⇔A₂ : A₁ ⇔ A₂) →
(∀ {x} → _⟶_ (B₁ at x) (B₂ at (Equivalence.to   A₁⇔A₂ ⟨\$⟩ x))) →
(∀ {y} → _⟶_ (B₂ at y) (B₁ at (Equivalence.from A₁⇔A₂ ⟨\$⟩ y))) →
Equivalence (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
equivalence {B₁ = B₁} {B₂} A₁⇔A₂ B-to B-from = record
{ to   = ⟶ B₂ (_⟨\$⟩_ (to   A₁⇔A₂)) B-to
; from = ⟶ B₁ (_⟨\$⟩_ (from A₁⇔A₂)) B-from
} where open Equivalence

private

subst-cong : ∀ {i a p} {I : Set i} {A : I → Set a}
(P : ∀ {i} → A i → A i → Set p) {i i′} {x y : A i}
(i≡i′ : P._≡_ i i′) →
P x y → P (P.subst A i≡i′ x) (P.subst A i≡i′ y)
subst-cong P P.refl p = p

equivalence-↞ :
∀ {a₁ a₂ b₁ b₁′ b₂ b₂′}
{A₁ : Set a₁} {A₂ : Set a₂}
(B₁ : I.Setoid A₁ b₁ b₁′) {B₂ : I.Setoid A₂ b₂ b₂′}
(A₁↞A₂ : A₁ ↞ A₂) →
(∀ {x} → Equivalence (B₁ at (LeftInverse.from A₁↞A₂ ⟨\$⟩ x))
(B₂ at x)) →
Equivalence (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
equivalence-↞ B₁ {B₂} A₁↞A₂ B₁⇔B₂ =
equivalence (LeftInverse.equivalence A₁↞A₂) B-to B-from
where
B-to : ∀ {x} → _⟶_ (B₁ at x) (B₂ at (LeftInverse.to A₁↞A₂ ⟨\$⟩ x))
B-to = record
{ _⟨\$⟩_ = λ x → Equivalence.to B₁⇔B₂ ⟨\$⟩
P.subst (I.Setoid.Carrier B₁)
(P.sym \$ LeftInverse.left-inverse-of A₁↞A₂ _)
x
; cong  = F.cong (Equivalence.to B₁⇔B₂) ∘
subst-cong (λ {x} → I.Setoid._≈_ B₁ {x} {x})
(P.sym (LeftInverse.left-inverse-of A₁↞A₂ _))
}

B-from : ∀ {y} → _⟶_ (B₂ at y) (B₁ at (LeftInverse.from A₁↞A₂ ⟨\$⟩ y))
B-from = Equivalence.from B₁⇔B₂

equivalence-↠ :
∀ {a₁ a₂ b₁ b₁′ b₂ b₂′}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : I.Setoid A₁ b₁ b₁′} (B₂ : I.Setoid A₂ b₂ b₂′)
(A₁↠A₂ : A₁ ↠ A₂) →
(∀ {x} → Equivalence (B₁ at x) (B₂ at (Surjection.to A₁↠A₂ ⟨\$⟩ x))) →
Equivalence (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
equivalence-↠ {B₁ = B₁} B₂ A₁↠A₂ B₁⇔B₂ =
equivalence (Surjection.equivalence A₁↠A₂) B-to B-from
where
B-to : ∀ {x} → _⟶_ (B₁ at x) (B₂ at (Surjection.to A₁↠A₂ ⟨\$⟩ x))
B-to = Equivalence.to B₁⇔B₂

B-from : ∀ {y} → _⟶_ (B₂ at y) (B₁ at (Surjection.from A₁↠A₂ ⟨\$⟩ y))
B-from = record
{ _⟨\$⟩_ = λ x → Equivalence.from B₁⇔B₂ ⟨\$⟩
P.subst (I.Setoid.Carrier B₂)
(P.sym \$ Surjection.right-inverse-of A₁↠A₂ _)
x
; cong  = F.cong (Equivalence.from B₁⇔B₂) ∘
subst-cong (λ {x} → I.Setoid._≈_ B₂ {x} {x})
(P.sym (Surjection.right-inverse-of A₁↠A₂ _))
}

⇔ : ∀ {a₁ a₂ b₁ b₂}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂}
(A₁⇔A₂ : A₁ ⇔ A₂) →
(∀ {x} → B₁ x → B₂ (Equivalence.to   A₁⇔A₂ ⟨\$⟩ x)) →
(∀ {y} → B₂ y → B₁ (Equivalence.from A₁⇔A₂ ⟨\$⟩ y)) →
Σ A₁ B₁ ⇔ Σ A₂ B₂
⇔ {B₁ = B₁} {B₂} A₁⇔A₂ B-to B-from =
Inverse.equivalence (Rel↔≡ {B = B₂}) ⟨∘⟩
equivalence A₁⇔A₂
(Inverse.to (H.≡↔≅ B₂) ⊚ P.→-to-⟶ B-to ⊚ Inverse.from (H.≡↔≅ B₁))
(Inverse.to (H.≡↔≅ B₁) ⊚ P.→-to-⟶ B-from ⊚ Inverse.from (H.≡↔≅ B₂))
⟨∘⟩
Eq.sym (Inverse.equivalence (Rel↔≡ {B = B₁}))
where
open Eq using () renaming (_∘_ to _⟨∘⟩_)
open F  using () renaming (_∘_ to _⊚_)

⇔-↠ : ∀ {a₁ a₂ b₁ b₂}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂}
(A₁↠A₂ : A₁ ↠ A₂) →
(∀ {x} → _⇔_ (B₁ x) (B₂ (Surjection.to A₁↠A₂ ⟨\$⟩ x))) →
_⇔_ (Σ A₁ B₁) (Σ A₂ B₂)
⇔-↠ {B₁ = B₁} {B₂} A₁↠A₂ B₁⇔B₂ =
Inverse.equivalence (Rel↔≡ {B = B₂}) ⟨∘⟩
equivalence-↠ (H.indexedSetoid B₂) A₁↠A₂
(λ {x} → Inverse.equivalence (H.≡↔≅ B₂) ⟨∘⟩
B₁⇔B₂ {x} ⟨∘⟩
Inverse.equivalence (Inv.sym (H.≡↔≅ B₁))) ⟨∘⟩
Eq.sym (Inverse.equivalence (Rel↔≡ {B = B₁}))
where open Eq using () renaming (_∘_ to _⟨∘⟩_)

injection :
∀ {a₁ a₂ b₁ b₁′ b₂ b₂′}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : I.Setoid A₁ b₁ b₁′} (B₂ : I.Setoid A₂ b₂ b₂′) →
(A₁↣A₂ : A₁ ↣ A₂) →
(∀ {x} → Injection (B₁ at x) (B₂ at (Injection.to A₁↣A₂ ⟨\$⟩ x))) →
Injection (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
injection {B₁ = B₁} B₂ A₁↣A₂ B₁↣B₂ = record
{ to        = to
; injective = inj
}
where
to = ⟶ B₂ (_⟨\$⟩_ (Injection.to A₁↣A₂)) (Injection.to B₁↣B₂)

inj : Injective to
inj (x , y) =
Injection.injective A₁↣A₂ x ,
lemma (Injection.injective A₁↣A₂ x) y
where
lemma :
∀ {x x′}
{y : I.Setoid.Carrier B₁ x} {y′ : I.Setoid.Carrier B₁ x′} →
P._≡_ x x′ →
(eq : I.Setoid._≈_ B₂ (Injection.to B₁↣B₂ ⟨\$⟩ y)
(Injection.to B₁↣B₂ ⟨\$⟩ y′)) →
I.Setoid._≈_ B₁ y y′
lemma P.refl = Injection.injective B₁↣B₂

↣ : ∀ {a₁ a₂ b₁ b₂}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂}
(A₁↣A₂ : A₁ ↣ A₂) →
(∀ {x} → B₁ x ↣ B₂ (Injection.to A₁↣A₂ ⟨\$⟩ x)) →
Σ A₁ B₁ ↣ Σ A₂ B₂
↣ {B₁ = B₁} {B₂} A₁↣A₂ B₁↣B₂ =
Inverse.injection (Rel↔≡ {B = B₂}) ⟨∘⟩
injection (H.indexedSetoid B₂) A₁↣A₂
(λ {x} → Inverse.injection (H.≡↔≅ B₂) ⟨∘⟩
B₁↣B₂ {x} ⟨∘⟩
Inverse.injection (Inv.sym (H.≡↔≅ B₁))) ⟨∘⟩
Inverse.injection (Inv.sym (Rel↔≡ {B = B₁}))
where open Inj using () renaming (_∘_ to _⟨∘⟩_)

left-inverse :
∀ {a₁ a₂ b₁ b₁′ b₂ b₂′}
{A₁ : Set a₁} {A₂ : Set a₂}
(B₁ : I.Setoid A₁ b₁ b₁′) {B₂ : I.Setoid A₂ b₂ b₂′} →
(A₁↞A₂ : A₁ ↞ A₂) →
(∀ {x} → LeftInverse (B₁ at (LeftInverse.from A₁↞A₂ ⟨\$⟩ x))
(B₂ at x)) →
LeftInverse (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
left-inverse B₁ {B₂} A₁↞A₂ B₁↞B₂ = record
{ to              = Equivalence.to   eq
; from            = Equivalence.from eq
; left-inverse-of = left
}
where
eq = equivalence-↞ B₁ A₁↞A₂ (LeftInverse.equivalence B₁↞B₂)

left : Equivalence.from eq LeftInverseOf Equivalence.to eq
left (x , y) =
LeftInverse.left-inverse-of A₁↞A₂ x ,
I.Setoid.trans B₁
(LeftInverse.left-inverse-of B₁↞B₂ _)
(lemma (P.sym (LeftInverse.left-inverse-of A₁↞A₂ x)))
where
lemma :
∀ {x x′ y} (eq : P._≡_ x x′) →
I.Setoid._≈_ B₁ (P.subst (I.Setoid.Carrier B₁) eq y) y
lemma P.refl = I.Setoid.refl B₁

↞ : ∀ {a₁ a₂ b₁ b₂}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂}
(A₁↞A₂ : A₁ ↞ A₂) →
(∀ {x} → B₁ (LeftInverse.from A₁↞A₂ ⟨\$⟩ x) ↞ B₂ x) →
Σ A₁ B₁ ↞ Σ A₂ B₂
↞ {B₁ = B₁} {B₂} A₁↞A₂ B₁↞B₂ =
Inverse.left-inverse (Rel↔≡ {B = B₂}) ⟨∘⟩
left-inverse (H.indexedSetoid B₁) A₁↞A₂
(λ {x} → Inverse.left-inverse (H.≡↔≅ B₂) ⟨∘⟩
B₁↞B₂ {x} ⟨∘⟩
Inverse.left-inverse (Inv.sym (H.≡↔≅ B₁))) ⟨∘⟩
Inverse.left-inverse (Inv.sym (Rel↔≡ {B = B₁}))
where open LeftInv using () renaming (_∘_ to _⟨∘⟩_)

surjection :
∀ {a₁ a₂ b₁ b₁′ b₂ b₂′}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : I.Setoid A₁ b₁ b₁′} (B₂ : I.Setoid A₂ b₂ b₂′) →
(A₁↠A₂ : A₁ ↠ A₂) →
(∀ {x} → Surjection (B₁ at x) (B₂ at (Surjection.to A₁↠A₂ ⟨\$⟩ x))) →
Surjection (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
surjection {B₁} B₂ A₁↠A₂ B₁↠B₂ = record
{ to         = Equivalence.to eq
; surjective = record
{ from             = Equivalence.from eq
; right-inverse-of = right
}
}
where
eq = equivalence-↠ B₂ A₁↠A₂ (Surjection.equivalence B₁↠B₂)

right : Equivalence.from eq RightInverseOf Equivalence.to eq
right (x , y) =
Surjection.right-inverse-of A₁↠A₂ x ,
I.Setoid.trans B₂
(Surjection.right-inverse-of B₁↠B₂ _)
(lemma (P.sym \$ Surjection.right-inverse-of A₁↠A₂ x))
where
lemma : ∀ {x x′ y} (eq : P._≡_ x x′) →
I.Setoid._≈_ B₂ (P.subst (I.Setoid.Carrier B₂) eq y) y
lemma P.refl = I.Setoid.refl B₂

↠ : ∀ {a₁ a₂ b₁ b₂}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂}
(A₁↠A₂ : A₁ ↠ A₂) →
(∀ {x} → B₁ x ↠ B₂ (Surjection.to A₁↠A₂ ⟨\$⟩ x)) →
Σ A₁ B₁ ↠ Σ A₂ B₂
↠ {B₁ = B₁} {B₂} A₁↠A₂ B₁↠B₂ =
Inverse.surjection (Rel↔≡ {B = B₂}) ⟨∘⟩
surjection (H.indexedSetoid B₂) A₁↠A₂
(λ {x} → Inverse.surjection (H.≡↔≅ B₂) ⟨∘⟩
B₁↠B₂ {x} ⟨∘⟩
Inverse.surjection (Inv.sym (H.≡↔≅ B₁))) ⟨∘⟩
Inverse.surjection (Inv.sym (Rel↔≡ {B = B₁}))
where open Surj using () renaming (_∘_ to _⟨∘⟩_)

inverse :
∀ {a₁ a₂ b₁ b₁′ b₂ b₂′}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : I.Setoid A₁ b₁ b₁′} (B₂ : I.Setoid A₂ b₂ b₂′) →
(A₁↔A₂ : A₁ ↔ A₂) →
(∀ {x} → Inverse (B₁ at x) (B₂ at (Inverse.to A₁↔A₂ ⟨\$⟩ x))) →
Inverse (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
inverse {B₁ = B₁} B₂ A₁↔A₂ B₁↔B₂ = record
{ to         = Surjection.to   surj
; from       = Surjection.from surj
; inverse-of = record
{ left-inverse-of  = left
; right-inverse-of = Surjection.right-inverse-of surj
}
}
where
surj = surjection B₂ (Inverse.surjection A₁↔A₂)
(Inverse.surjection B₁↔B₂)

left : Surjection.from surj LeftInverseOf Surjection.to surj
left (x , y) =
Inverse.left-inverse-of A₁↔A₂ x ,
I.Setoid.trans B₁
(lemma (P.sym (Inverse.left-inverse-of A₁↔A₂ x))
(P.sym (Inverse.right-inverse-of A₁↔A₂
(Inverse.to A₁↔A₂ ⟨\$⟩ x))))
(Inverse.left-inverse-of B₁↔B₂ y)
where
lemma :
∀ {x x′ y} → P._≡_ x x′ →
(eq : P._≡_ (Inverse.to A₁↔A₂ ⟨\$⟩ x) (Inverse.to A₁↔A₂ ⟨\$⟩ x′)) →
I.Setoid._≈_ B₁
(Inverse.from B₁↔B₂ ⟨\$⟩ P.subst (I.Setoid.Carrier B₂) eq y)
(Inverse.from B₁↔B₂ ⟨\$⟩ y)
lemma P.refl P.refl = I.Setoid.refl B₁

↔ : ∀ {a₁ a₂ b₁ b₂}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂}
(A₁↔A₂ : A₁ ↔ A₂) →
(∀ {x} → B₁ x ↔ B₂ (Inverse.to A₁↔A₂ ⟨\$⟩ x)) →
Σ A₁ B₁ ↔ Σ A₂ B₂
↔ {B₁ = B₁} {B₂} A₁↔A₂ B₁↔B₂ =
Rel↔≡ {B = B₂} ⟨∘⟩
inverse (H.indexedSetoid B₂) A₁↔A₂
(λ {x} → H.≡↔≅ B₂ ⟨∘⟩ B₁↔B₂ {x} ⟨∘⟩ Inv.sym (H.≡↔≅ B₁)) ⟨∘⟩
Inv.sym (Rel↔≡ {B = B₁})
where open Inv using () renaming (_∘_ to _⟨∘⟩_)

private

swap-coercions :
∀ {k a₁ a₂ b₁ b₂}
{A₁ : Set a₁} {A₂ : Set a₂} {B₁ : A₁ → Set b₁} (B₂ : A₂ → Set b₂)
(A₁↔A₂ : _↔_ A₁ A₂) →
(∀ {x} → B₁ x ∼[ k ] B₂ (Inverse.to A₁↔A₂ ⟨\$⟩ x)) →
∀ {x} → B₁ (Inverse.from A₁↔A₂ ⟨\$⟩ x) ∼[ k ] B₂ x
swap-coercions {k} {B₁ = B₁} B₂ A₁↔A₂ eq {x} =
B₁ (Inverse.from A₁↔A₂ ⟨\$⟩ x)                         ∼⟨ eq ⟩
B₂ (Inverse.to A₁↔A₂ ⟨\$⟩ (Inverse.from A₁↔A₂ ⟨\$⟩ x))  ↔⟨ B.Setoid.reflexive (Related.setoid Related.bijection _)
(P.cong B₂ \$ Inverse.right-inverse-of A₁↔A₂ x) ⟩
B₂ x                                                  ∎
where open Related.EquationalReasoning

cong : ∀ {k a₁ a₂ b₁ b₂}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂}
(A₁↔A₂ : _↔_ A₁ A₂) →
(∀ {x} → B₁ x ∼[ k ] B₂ (Inverse.to A₁↔A₂ ⟨\$⟩ x)) →
Σ A₁ B₁ ∼[ k ] Σ A₂ B₂
cong {Related.implication}                   = λ A₁↔A₂ → Prod.map (_⟨\$⟩_ (Inverse.to A₁↔A₂))
cong {Related.reverse-implication} {B₂ = B₂} = λ A₁↔A₂ B₁←B₂ → lam (Prod.map (_⟨\$⟩_ (Inverse.from A₁↔A₂))
(app-← (swap-coercions B₂ A₁↔A₂ B₁←B₂)))
cong {Related.equivalence}                   = ⇔-↠ ∘ Inverse.surjection
cong {Related.injection}                     = ↣ ∘ Inverse.injection
cong {Related.reverse-injection}   {B₂ = B₂} = λ A₁↔A₂ B₁↢B₂ → lam (↣ (Inverse.injection (Inv.sym A₁↔A₂))
(app-↢ (swap-coercions B₂ A₁↔A₂ B₁↢B₂)))
cong {Related.left-inverse}                  = λ A₁↔A₂ → ↞ (Inverse.left-inverse A₁↔A₂) ∘ swap-coercions _ A₁↔A₂
cong {Related.surjection}                    = ↠ ∘ Inverse.surjection
cong {Related.bijection}                     = ↔
```