```{-# OPTIONS --without-K #-}

open import lib.Base
open import lib.PathGroupoid
open import lib.PathFunctor
open import lib.NType

module lib.Equivalences where

{-
We use the half-adjoint definition of equivalences (but this fact should be
invisible to the user of the library). The constructor of the type of
equivalences is [equiv], it takes two maps and the two proofs that the composites
are equal: [equiv to from to-from from-to]

The type of equivalences between two types [A] and [B] can be written either
[A ≃ B] or [Equiv A B].

Given an equivalence [e] : [A ≃ B], you can extract the two maps as follows:
[–> e] : [A → B] and [<– e] : [B → A] (the dash is an en dash)
The proofs that the composites are the identities are [<–-inv-l] and [<–-inv-r].

The identity equivalence on [A] is [ide A], the composition of two equivalences
is [_∘e_] (function composition order) and the inverse of an equivalence is [_⁻¹]
-}

module _ {i} {j} {A : Type i} {B : Type j} where

record is-equiv (f : A → B) : Type (lmax i j)
where
field
g : B → A
f-g : (b : B) → f (g b) == b
g-f : (a : A) → g (f a) == a
adj : (a : A) → ap f (g-f a) == f-g (f a)

{-
In order to prove that something is an equivalence, you have to give an inverse
and a proof that it’s an inverse (you don’t need the adj part).
[is-eq] is a very, very bad name.
-}
is-eq : (f : A → B)
(g : B → A) (f-g : (b : B) → f (g b) == b)
(g-f : (a : A) → g (f a) == a) → is-equiv f
is-eq f g f-g g-f =
record {g = g; f-g = f-g'; g-f = g-f; adj = adj} where
f-g' : (b : B) → f (g b) == b
f-g' b = ! (ap (f ∘ g) (f-g b)) ∙ ap f (g-f (g b)) ∙ f-g b

adj : (a : A) → ap f (g-f a) == f-g' (f a)
ap f (g-f a)
=⟨ ! (!-inv-l (ap (f ∘ g) (f-g (f a)))) |in-ctx (λ q → q ∙ ap f (g-f a)) ⟩
(! (ap (f ∘ g) (f-g (f a))) ∙ ap (f ∘ g) (f-g (f a))) ∙ ap f (g-f a)
=⟨ ∙-assoc (! (ap (f ∘ g) (f-g (f a)))) (ap (f ∘ g) (f-g (f a))) _ ⟩
! (ap (f ∘ g) (f-g (f a))) ∙ ap (f ∘ g) (f-g (f a)) ∙ ap f (g-f a)
=⟨ lemma |in-ctx (λ q → ! (ap (f ∘ g) (f-g (f a))) ∙ q) ⟩
! (ap (f ∘ g) (f-g (f a))) ∙ ap f (g-f (g (f a))) ∙ f-g (f a) ∎
where
lemma : ap (f ∘ g) (f-g (f a)) ∙ ap f (g-f a)
== ap f (g-f (g (f a))) ∙ f-g (f a)
lemma =
ap (f ∘ g) (f-g (f a)) ∙ ap f (g-f a)
=⟨ htpy-natural-toid f-g (f a) |in-ctx (λ q → q ∙ ap f (g-f a)) ⟩
f-g (f (g (f a))) ∙ ap f (g-f a)
=⟨ ! (ap-idf (ap f (g-f a))) |in-ctx (λ q → f-g (f (g (f a))) ∙ q) ⟩
f-g (f (g (f a))) ∙ ap (idf B) (ap f (g-f a))
=⟨ ! (htpy-natural f-g (ap f (g-f a))) ⟩
ap (f ∘ g) (ap f (g-f a)) ∙ f-g (f a)
=⟨ ap-∘ f g (ap f (g-f a)) |in-ctx (λ q → q ∙ f-g (f a)) ⟩
ap f (ap g (ap f (g-f a))) ∙ f-g (f a)
=⟨ ∘-ap g f (g-f a) ∙ htpy-natural-toid g-f a
|in-ctx (λ q → ap f q ∙ f-g (f a)) ⟩
ap f (g-f (g (f a))) ∙ f-g (f a) ∎

infix 4 _≃_

_≃_ : ∀ {i j} (A : Type i) (B : Type j) → Type (lmax i j)
A ≃ B = Σ (A → B) is-equiv

Equiv = _≃_

module _ {i} {j} {A : Type i} {B : Type j} where

equiv : (f : A → B) (g : B → A) (f-g : (b : B) → f (g b) == b)
(g-f : (a : A) → g (f a) == a) → A ≃ B
equiv f g f-g g-f = (f , is-eq f g f-g g-f)

–> : (e : A ≃ B) → (A → B)
–> e = fst e

<– : (e : A ≃ B) → (B → A)
<– e = is-equiv.g (snd e)

<–-inv-l : (e : A ≃ B) (a : A)
→ (<– e (–> e a) == a)
<–-inv-l e a = is-equiv.g-f (snd e) a

<–-inv-r : (e : A ≃ B) (b : B)
→ (–> e (<– e b) == b)
<–-inv-r e b = is-equiv.f-g (snd e) b

-- Equivalences are "injective"
equiv-inj : (e : A ≃ B) {x y : A}
→ (–> e x == –> e y → x == y)
equiv-inj e {x} {y} p = ! (<–-inv-l e x) ∙ ap (<– e) p ∙ <–-inv-l e y

idf-is-equiv : ∀ {i} (A : Type i) → is-equiv (idf A)
idf-is-equiv A = is-eq _ (idf A) (λ _ → idp) (λ _ → idp)

ide : ∀ {i} (A : Type i) → A ≃ A
ide A = equiv (idf A) (idf A) (λ _ → idp) (λ _ → idp)

infixr 4 _∘e_
infixr 4 _∘ise_

_∘e_ : ∀ {i j k} {A : Type i} {B : Type j} {C : Type k}
→ B ≃ C → A ≃ B → A ≃ C
e1 ∘e e2 = equiv (–> e1 ∘ –> e2) (<– e2 ∘ <– e1)
(λ c → –> e1 (–> e2 (<– e2 (<– e1 c)))
=⟨ <–-inv-r e2 (<– e1 c) |in-ctx (–> e1) ⟩
–> e1 (<– e1 c)  =⟨ <–-inv-r e1 c ⟩
c ∎)
(λ a → <– e2 (<– e1 (–> e1 (–> e2 a)))
=⟨ <–-inv-l e1 (–> e2 a) |in-ctx (<– e2) ⟩
<– e2 (–> e2 a)  =⟨ <–-inv-l e2 a ⟩
a ∎)

_∘ise_ : ∀ {i j k} {A : Type i} {B : Type j} {C : Type k}
{f : A → B} {g : B → C}
→ is-equiv g → is-equiv f → is-equiv (g ∘ f)
i1 ∘ise i2 = snd ((_ , i1) ∘e (_ , i2))

_⁻¹ : ∀ {i j} {A : Type i} {B : Type j} → (A ≃ B) → (B ≃ A)
e ⁻¹ = equiv (<– e) (–> e) (<–-inv-l e) (<–-inv-r e)

{- Equational reasoning for equivalences -}
infix  2 _≃∎
infixr 2 _≃⟨_⟩_

_≃⟨_⟩_ : ∀ {i j k} (A : Type i) {B : Type j} {C : Type k} → A ≃ B → B ≃ C → A ≃ C
A ≃⟨ u ⟩ v = v ∘e u

_≃∎ : ∀ {i} (A : Type i) → A ≃ A
_≃∎ = ide

{- lifting is an equivalence -}
lift-equiv : ∀ {i j} {A : Type i} → Lift {j = j} A ≃ A
lift-equiv = equiv lower lift (λ _ → idp) (λ _ → idp)

{- Any contractible type is equivalent to (all liftings of) the unit type -}
module _ {i} {A : Type i} (h : is-contr A) where
contr-equiv-Unit : A ≃ Unit
contr-equiv-Unit = equiv (λ _ → unit) (λ _ → fst h) (λ _ → idp) (snd h)

contr-equiv-LiftUnit : ∀ {j} → A ≃ Lift {j = j} Unit
contr-equiv-LiftUnit = lift-equiv ⁻¹ ∘e contr-equiv-Unit

{- An equivalence induces an equivalence on the path spaces -}
module _ {i j} {A : Type i} {B : Type j} (e : A ≃ B) where

private
abstract
left-inverse : {x y : A} (p : x == y) → equiv-inj e (ap (–> e) p) == p
left-inverse idp = !-inv-l (<–-inv-l e _)

right-inverse : {x y : A} (p : –> e x == –> e y)
→ ap (–> e) (equiv-inj e p) == p
right-inverse {x} {y} p =
ap f (! (g-f x) ∙ ap g p ∙ (g-f y))
=⟨ ap-∙ f (! (g-f x)) (ap g p ∙ (g-f y)) ⟩
ap f (! (g-f x)) ∙ ap f (ap g p ∙ (g-f y))
=⟨ ap-∙ f (ap g p) (g-f y) |in-ctx (λ q →  ap f (! (g-f x)) ∙ q) ⟩
ap f (! (g-f x)) ∙ ap f (ap g p) ∙ ap f (g-f y)
=⟨ ∘-ap f g p |in-ctx (λ q → ap f (! (g-f x)) ∙ q ∙ ap f (g-f y)) ⟩
ap f (! (g-f x)) ∙ ap (f ∘ g) p ∙ ap f (g-f y)
=⟨ adj y |in-ctx (λ q →  ap f (! (g-f x)) ∙ ap (f ∘ g) p ∙ q) ⟩
ap f (! (g-f x)) ∙ ap (f ∘ g) p ∙ (f-g (f y))
=⟨ ap-! f (g-f x) |in-ctx (λ q → q ∙ ap (f ∘ g) p ∙ (f-g (f y))) ⟩
! (ap f (g-f x)) ∙ ap (f ∘ g) p ∙ (f-g (f y))
=⟨ adj x |in-ctx (λ q →  ! q ∙ ap (f ∘ g) p ∙ (f-g (f y))) ⟩
! (f-g (f x)) ∙ ap (f ∘ g) p ∙ (f-g (f y))
=⟨ htpy-natural f-g p |in-ctx (λ q →  ! (f-g (f x)) ∙ q) ⟩
! (f-g (f x)) ∙ (f-g (f x)) ∙ ap (idf B) p
=⟨ ! (∙-assoc (! (f-g (f x))) (f-g (f x)) (ap (idf B) p))
∙ ap (λ q → q ∙ ap (idf B) p) (!-inv-l (f-g (f x))) ∙ ap-idf p ⟩
p ∎
where f : A → B
f = fst e

open is-equiv (snd e)

equiv-ap : (x y : A) → (x == y) ≃ (–> e x == –> e y)
equiv-ap x y = equiv (ap (–> e)) (equiv-inj e) right-inverse left-inverse

{- Equivalent types have the same truncation level -}
equiv-preserves-level : ∀ {i j} {A : Type i} {B : Type j} {n : ℕ₋₂} (e : A ≃ B)
→ (has-level n A → has-level n B)
equiv-preserves-level {n = ⟨-2⟩} e (x , p) =
(–> e x , (λ y → ap (–> e) (p _) ∙ <–-inv-r e y))
equiv-preserves-level {n = S n} e c = λ x y →
equiv-preserves-level (equiv-ap (e ⁻¹) x y ⁻¹) (c (<– e x) (<– e y))

{- This is a collection of type equivalences involving basic type formers.
We exclude Empty since Π₁-Empty requires λ=.
-}
module _ {j} {B : Unit → Type j} where
Σ₁-Unit : Σ Unit B ≃ B unit
Σ₁-Unit = equiv (λ {(unit , b) → b}) (λ b → (unit , b)) (λ _ → idp) (λ _ → idp)

Π₁-Unit : Π Unit B ≃ B unit
Π₁-Unit = equiv (λ f → f unit) (λ b _ → b) (λ _ → idp) (λ _ → idp)

module _ {i} {A : Type i} where
Σ₂-Unit : Σ A (λ _ → Unit) ≃ A
Σ₂-Unit = equiv fst (λ a → (a , unit)) (λ _ → idp) (λ _ → idp)

Π₂-Unit : Π A (λ _ → Unit) ≃ Unit
Π₂-Unit = equiv (λ _ → unit) (λ _ _ → unit) (λ _ → idp) (λ _ → idp)

module _ {i j k} {A : Type i} {B : A → Type j} {C : (a : A) → B a → Type k} where
Σ-assoc : Σ (Σ A B) (uncurry C) ≃ Σ A (λ a → Σ (B a) (C a))
Σ-assoc = equiv (λ {((a , b) , c) → (a , (b , c))})
(λ {(a , (b , c)) → ((a , b) , c)}) (λ _ → idp) (λ _ → idp)

curry-equiv : Π (Σ A B) (uncurry C) ≃ Π A (λ a → Π (B a) (C a))
curry-equiv = equiv curry uncurry (λ _ → idp) (λ _ → idp)

{- The type-theoretic axiom of choice -}
choice : Π A (λ a → Σ (B a) (λ b → C a b)) ≃ Σ (Π A B) (λ g → Π A (λ a → C a (g a)))
choice = equiv f g (λ _ → idp) (λ _ → idp)
where f = λ c → ((λ a → fst (c a)) , (λ a → snd (c a)))
g = λ d → (λ a → (fst d a , snd d a))

{- Pre- and post- concatenation are equivalences -}
module _ {i} {A : Type i} {x y z : A} where

pre∙-is-equiv : (p : x == y) → is-equiv (λ (q : y == z) → p ∙ q)
pre∙-is-equiv p = is-eq (λ q → p ∙ q) (λ r → ! p ∙ r) f-g g-f
where f-g : ∀ r → p ∙ ! p ∙ r == r
f-g r = ! (∙-assoc p (! p) r) ∙ ap (λ s → s ∙ r) (!-inv-r p)

g-f : ∀ q → ! p ∙ p ∙ q == q
g-f q = ! (∙-assoc (! p) p q) ∙ ap (λ s → s ∙ q) (!-inv-l p)

post∙-is-equiv : (p : y == z) → is-equiv (λ (q : x == y) → q ∙ p)
post∙-is-equiv p = is-eq (λ q → q ∙ p) (λ r → r ∙ ! p) f-g g-f
where f-g : ∀ r → (r ∙ ! p) ∙ p == r
f-g r = ∙-assoc r (! p) p ∙ ap (λ s → r ∙ s) (!-inv-l p) ∙ ∙-unit-r r

g-f : ∀ q → (q ∙ p) ∙ ! p == q
g-f q = ∙-assoc q p (! p) ∙ ap (λ s → q ∙ s) (!-inv-r p) ∙ ∙-unit-r q
```