# Equality of dependent pair types

Content created by Fredrik Bakke, Egbert Rijke, Jonathan Prieto-Cubides, maybemabeline, Julian KG, fernabnor and louismntnu.

Created on 2022-02-07.

module foundation-core.equality-dependent-pair-types where

Imports
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import foundation-core.dependent-identifications
open import foundation-core.equivalences
open import foundation-core.function-types
open import foundation-core.homotopies
open import foundation-core.identity-types
open import foundation-core.transport-along-identifications


## Idea

An identification (pair x y) ＝ (pair x' y') in a dependent pair type Σ A B is equivalently described as a pair pair α β consisting of an identification α : x ＝ x' and an identification β : (tr B α y) ＝ y'.

## Definition

module _
{l1 l2 : Level} {A : UU l1} {B : A → UU l2}
where

Eq-Σ : (s t : Σ A B) → UU (l1 ⊔ l2)
Eq-Σ s t =
Σ (pr1 s ＝ pr1 t) (λ α → dependent-identification B α (pr2 s) (pr2 t))


## Properties

### The type Id s t is equivalent to Eq-Σ s t for any s t : Σ A B

  refl-Eq-Σ : (s : Σ A B) → Eq-Σ s s
pr1 (refl-Eq-Σ (pair a b)) = refl
pr2 (refl-Eq-Σ (pair a b)) = refl

pair-eq-Σ : {s t : Σ A B} → s ＝ t → Eq-Σ s t
pair-eq-Σ {s} refl = refl-Eq-Σ s

eq-pair-eq-base :
{x y : A} {s : B x} (p : x ＝ y) → (x , s) ＝ (y , tr B p s)
eq-pair-eq-base refl = refl

eq-pair-eq-base' :
{x y : A} {t : B y} (p : x ＝ y) → (x , tr B (inv p) t) ＝ (y , t)
eq-pair-eq-base' refl = refl

eq-pair-eq-fiber :
{x : A} {s t : B x} → s ＝ t → (x , s) ＝ (x , t)
eq-pair-eq-fiber {x} = ap {B = Σ A B} (pair x)

eq-pair-Σ :
{s t : Σ A B}
(α : pr1 s ＝ pr1 t) →
dependent-identification B α (pr2 s) (pr2 t) → s ＝ t
eq-pair-Σ refl = eq-pair-eq-fiber

eq-pair-Σ' : {s t : Σ A B} → Eq-Σ s t → s ＝ t
eq-pair-Σ' p = eq-pair-Σ (pr1 p) (pr2 p)

ap-pr1-eq-pair-eq-fiber :
{x : A} {s t : B x} (p : s ＝ t) → ap pr1 (eq-pair-eq-fiber p) ＝ refl
ap-pr1-eq-pair-eq-fiber refl = refl

is-retraction-pair-eq-Σ :
(s t : Σ A B) → pair-eq-Σ {s} {t} ∘ eq-pair-Σ' {s} {t} ~ id {A = Eq-Σ s t}
is-retraction-pair-eq-Σ (pair x y) (pair .x .y) (pair refl refl) = refl

is-section-pair-eq-Σ :
(s t : Σ A B) → ((eq-pair-Σ' {s} {t}) ∘ (pair-eq-Σ {s} {t})) ~ id
is-section-pair-eq-Σ (pair x y) .(pair x y) refl = refl

abstract
is-equiv-eq-pair-Σ : (s t : Σ A B) → is-equiv (eq-pair-Σ' {s} {t})
is-equiv-eq-pair-Σ s t =
is-equiv-is-invertible
( pair-eq-Σ)
( is-section-pair-eq-Σ s t)
( is-retraction-pair-eq-Σ s t)

equiv-eq-pair-Σ : (s t : Σ A B) → Eq-Σ s t ≃ (s ＝ t)
pr1 (equiv-eq-pair-Σ s t) = eq-pair-Σ'
pr2 (equiv-eq-pair-Σ s t) = is-equiv-eq-pair-Σ s t

abstract
is-equiv-pair-eq-Σ : (s t : Σ A B) → is-equiv (pair-eq-Σ {s} {t})
is-equiv-pair-eq-Σ s t =
is-equiv-is-invertible
( eq-pair-Σ')
( is-retraction-pair-eq-Σ s t)
( is-section-pair-eq-Σ s t)

equiv-pair-eq-Σ : (s t : Σ A B) → (s ＝ t) ≃ Eq-Σ s t
pr1 (equiv-pair-eq-Σ s t) = pair-eq-Σ
pr2 (equiv-pair-eq-Σ s t) = is-equiv-pair-eq-Σ s t

η-pair : (t : Σ A B) → (pair (pr1 t) (pr2 t)) ＝ t
η-pair t = refl

eq-base-eq-pair-Σ : {s t : Σ A B} → (s ＝ t) → (pr1 s ＝ pr1 t)
eq-base-eq-pair-Σ p = pr1 (pair-eq-Σ p)

dependent-eq-family-eq-pair-Σ :
{s t : Σ A B} → (p : s ＝ t) →
dependent-identification B (eq-base-eq-pair-Σ p) (pr2 s) (pr2 t)
dependent-eq-family-eq-pair-Σ p = pr2 (pair-eq-Σ p)


### Lifting equality to the total space

module _
{l1 l2 : Level} {A : UU l1} {B : A → UU l2}
where

lift-eq-Σ :
{x y : A} (p : x ＝ y) (b : B x) → (pair x b) ＝ (pair y (tr B p b))
lift-eq-Σ refl b = refl


### Transport in a family of dependent pair types

tr-Σ :
{l1 l2 l3 : Level} {A : UU l1} {a0 a1 : A} {B : A → UU l2}
(C : (x : A) → B x → UU l3) (p : a0 ＝ a1) (z : Σ (B a0) (λ x → C a0 x)) →
tr (λ a → (Σ (B a) (C a))) p z ＝
pair (tr B p (pr1 z)) (tr (ind-Σ C) (eq-pair-Σ p refl) (pr2 z))
tr-Σ C refl z = refl


### Transport in a family over a dependent pair type

tr-eq-pair-Σ :
{l1 l2 l3 : Level} {A : UU l1} {a0 a1 : A}
{B : A → UU l2} {b0 : B a0} {b1 : B a1} (C : (Σ A B) → UU l3)
(p : a0 ＝ a1) (q : dependent-identification B p b0 b1) (u : C (a0 , b0)) →
tr C (eq-pair-Σ p q) u ＝
tr (λ x → C (a1 , x)) q (tr C (eq-pair-Σ p refl) u)
tr-eq-pair-Σ C refl refl u = refl