```------------------------------------------------------------------------
-- A universe which includes several kinds of functions
------------------------------------------------------------------------

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

open import Equality

module Function-universe
{reflexive} (eq : ∀ {a p} → Equality-with-J a p reflexive) where

open import Bijection eq as Bijection using (_↔_; module _↔_)
open Derived-definitions-and-properties eq
open import Equality.Decidable-UIP eq
open import Equality.Decision-procedures eq
open import Equivalence eq as Eq using (_≃_; module _≃_)
open import H-level eq as H-level
open import H-level.Closure eq
open import Injection eq as Injection using (_↣_; module _↣_; Injective)
open import Logical-equivalence using (_⇔_; module _⇔_)
open import Nat eq hiding (_≟_)
open import Preimage eq using (_⁻¹_)
open import Prelude as P hiding (id) renaming (_∘_ to _⊚_)
open import Surjection eq as Surjection using (_↠_; module _↠_)

------------------------------------------------------------------------
-- The universe

-- The universe includes implications, logical equivalences,
-- injections, surjections, bijections and equivalences.

data Kind : Set where
implication
logical-equivalence
injection
surjection
bijection
equivalence : Kind

-- The interpretation of the universe.

infix 0 _↝[_]_

_↝[_]_ : ∀ {ℓ₁ ℓ₂} → Set ℓ₁ → Kind → Set ℓ₂ → Set _
A ↝[ implication         ] B = A → B
A ↝[ logical-equivalence ] B = A ⇔ B
A ↝[ injection           ] B = A ↣ B
A ↝[ surjection          ] B = A ↠ B
A ↝[ bijection           ] B = A ↔ B
A ↝[ equivalence         ] B = A ≃ B

-- Bijections can be converted to all kinds of functions.

from-bijection : ∀ {k a b} {A : Set a} {B : Set b} →
A ↔ B → A ↝[ k ] B
from-bijection {implication}         = _↔_.to
from-bijection {logical-equivalence} = _↔_.logical-equivalence
from-bijection {injection}           = _↔_.injection
from-bijection {surjection}          = _↔_.surjection
from-bijection {bijection}           = P.id
from-bijection {equivalence}         = Eq.↔⇒≃

-- Equivalences can be converted to all kinds of functions.

from-equivalence : ∀ {k a b} {A : Set a} {B : Set b} →
A ≃ B → A ↝[ k ] B
from-equivalence {implication}         = _≃_.to
from-equivalence {logical-equivalence} = _≃_.logical-equivalence
from-equivalence {injection}           = _≃_.injection
from-equivalence {surjection}          = _≃_.surjection
from-equivalence {bijection}           = _≃_.bijection
from-equivalence {equivalence}         = P.id

-- All kinds of functions can be converted to implications.

to-implication : ∀ {k a b} {A : Set a} {B : Set b} →
A ↝[ k ] B → A → B
to-implication {implication}         = P.id
to-implication {logical-equivalence} = _⇔_.to
to-implication {injection}           = _↣_.to
to-implication {surjection}          = _↠_.to
to-implication {bijection}           = _↔_.to
to-implication {equivalence}         = _≃_.to

------------------------------------------------------------------------
-- A sub-universe of symmetric kinds of functions

data Symmetric-kind : Set where
logical-equivalence bijection equivalence : Symmetric-kind

⌊_⌋-sym : Symmetric-kind → Kind
⌊ logical-equivalence ⌋-sym = logical-equivalence
⌊ bijection           ⌋-sym = bijection
⌊ equivalence         ⌋-sym = equivalence

inverse : ∀ {k a b} {A : Set a} {B : Set b} →
A ↝[ ⌊ k ⌋-sym ] B → B ↝[ ⌊ k ⌋-sym ] A
inverse {logical-equivalence} = Logical-equivalence.inverse
inverse {bijection}           = Bijection.inverse
inverse {equivalence}         = Eq.inverse

------------------------------------------------------------------------
-- A sub-universe of isomorphisms

data Isomorphism-kind : Set where
bijection equivalence : Isomorphism-kind

⌊_⌋-iso : Isomorphism-kind → Kind
⌊ bijection   ⌋-iso = bijection
⌊ equivalence ⌋-iso = equivalence

infix 0 _↔[_]_

_↔[_]_ : ∀ {ℓ₁ ℓ₂} → Set ℓ₁ → Isomorphism-kind → Set ℓ₂ → Set _
A ↔[ k ] B = A ↝[ ⌊ k ⌋-iso ] B

from-isomorphism : ∀ {k₁ k₂ a b} {A : Set a} {B : Set b} →
A ↔[ k₁ ] B → A ↝[ k₂ ] B
from-isomorphism {bijection}   = from-bijection
from-isomorphism {equivalence} = from-equivalence

-- Lemma: to-implication after from-isomorphism is the same as
-- to-implication.

to-implication∘from-isomorphism :
∀ {a b} {A : Set a} {B : Set b} k₁ k₂ {A↔B : A ↔[ k₁ ] B} →
to-implication A↔B ≡
to-implication (from-isomorphism {k₂ = k₂} A↔B)
to-implication∘from-isomorphism {A = A} {B} = t∘f
where
t∘f : ∀ k₁ k₂ {A↔B : A ↔[ k₁ ] B} →
to-implication A↔B ≡
to-implication (from-isomorphism {k₂ = k₂} A↔B)
t∘f bijection   implication         = refl _
t∘f bijection   logical-equivalence = refl _
t∘f bijection   injection           = refl _
t∘f bijection   surjection          = refl _
t∘f bijection   bijection           = refl _
t∘f bijection   equivalence         = refl _
t∘f equivalence implication         = refl _
t∘f equivalence logical-equivalence = refl _
t∘f equivalence injection           = refl _
t∘f equivalence surjection          = refl _
t∘f equivalence bijection           = refl _
t∘f equivalence equivalence         = refl _

------------------------------------------------------------------------
-- Preorder

-- All the different kinds of functions form preorders.

-- Composition.

infixr 9 _∘_

_∘_ : ∀ {k a b c} {A : Set a} {B : Set b} {C : Set c} →
B ↝[ k ] C → A ↝[ k ] B → A ↝[ k ] C
_∘_ {implication}         = λ f g → f ⊚ g
_∘_ {logical-equivalence} = Logical-equivalence._∘_
_∘_ {injection}           = Injection._∘_
_∘_ {surjection}          = Surjection._∘_
_∘_ {bijection}           = Bijection._∘_
_∘_ {equivalence}         = Eq._∘_

-- Identity.

id : ∀ {k a} {A : Set a} → A ↝[ k ] A
id {implication}         = P.id
id {logical-equivalence} = Logical-equivalence.id
id {injection}           = Injection.id
id {surjection}          = Surjection.id
id {bijection}           = Bijection.id
id {equivalence}         = Eq.id

-- "Equational" reasoning combinators.

infix  -1 finally-↝ finally-↔
infix  -1 _□
infixr -2 _↝⟨_⟩_ _↔⟨_⟩_ _↔⟨⟩_
infix  -3 \$⟨_⟩_

_↝⟨_⟩_ : ∀ {k a b c} (A : Set a) {B : Set b} {C : Set c} →
A ↝[ k ] B → B ↝[ k ] C → A ↝[ k ] C
_ ↝⟨ A↝B ⟩ B↝C = B↝C ∘ A↝B

_↔⟨_⟩_ : ∀ {k₁ k₂ a b c} (A : Set a) {B : Set b} {C : Set c} →
A ↔[ k₁ ] B → B ↝[ k₂ ] C → A ↝[ k₂ ] C
_ ↔⟨ A↔B ⟩ B↝C = _ ↝⟨ from-isomorphism A↔B ⟩ B↝C

_↔⟨⟩_ : ∀ {k a b} (A : Set a) {B : Set b} →
A ↝[ k ] B → A ↝[ k ] B
_ ↔⟨⟩ A↝B = A↝B

_□ : ∀ {k a} (A : Set a) → A ↝[ k ] A
A □ = id

finally-↝ : ∀ {k a b} (A : Set a) (B : Set b) →
A ↝[ k ] B → A ↝[ k ] B
finally-↝ _ _ A↝B = A↝B

finally-↔ : ∀ {k₁ k₂ a b} (A : Set a) (B : Set b) →
A ↔[ k₁ ] B → A ↝[ k₂ ] B
finally-↔ _ _ A↔B = from-isomorphism A↔B

syntax finally-↝ A B A↝B = A ↝⟨ A↝B ⟩□ B □
syntax finally-↔ A B A↔B = A ↔⟨ A↔B ⟩□ B □

\$⟨_⟩_ : ∀ {k a b} {A : Set a} {B : Set b} →
A → A ↝[ k ] B → B
\$⟨ a ⟩ A↝B = to-implication A↝B a

-- Lemma: to-implication maps id to the identity function.

to-implication-id :
∀ {a} {A : Set a} k →
to-implication {k = k} id ≡ id {A = A}
to-implication-id implication         = refl _
to-implication-id logical-equivalence = refl _
to-implication-id injection           = refl _
to-implication-id surjection          = refl _
to-implication-id bijection           = refl _
to-implication-id equivalence         = refl _

-- Lemma: to-implication is homomorphic with respect to _∘_.

to-implication-∘ :
∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
(k : Kind) {f : A ↝[ k ] B} {g : B ↝[ k ] C} →
to-implication (g ∘ f) ≡ to-implication g ∘ to-implication f
to-implication-∘ implication         = refl _
to-implication-∘ logical-equivalence = refl _
to-implication-∘ injection           = refl _
to-implication-∘ surjection          = refl _
to-implication-∘ bijection           = refl _
to-implication-∘ equivalence         = refl _

-- Lemma: to-implication maps inverse id to the identity function.

to-implication-inverse-id :
∀ {a} {A : Set a} k →
to-implication (inverse {k = k} id) ≡ id {A = A}
to-implication-inverse-id logical-equivalence = refl _
to-implication-inverse-id bijection           = refl _
to-implication-inverse-id equivalence         = refl _

------------------------------------------------------------------------
-- Lots of properties
------------------------------------------------------------------------

-- Properties of the form A ↝[ k ] B, for arbitrary k, are only stated
-- for bijections or equivalences; converting to the other forms is
-- easy.

------------------------------------------------------------------------
-- Equalities can be converted to all kinds of functions

≡⇒↝ : ∀ k {ℓ} {A B : Set ℓ} → A ≡ B → A ↝[ k ] B
≡⇒↝ k = elim (λ {A B} _ → A ↝[ k ] B) (λ _ → id)

abstract

-- Some lemmas that can be used to manipulate expressions involving
-- ≡⇒↝ and refl/sym/trans.

≡⇒↝-refl : ∀ {k a} {A : Set a} →
≡⇒↝ k (refl A) ≡ id
≡⇒↝-refl {k} = elim-refl (λ {A B} _ → A ↝[ k ] B) _

≡⇒↝-sym : ∀ k {ℓ} {A B : Set ℓ} {eq : A ≡ B} →
to-implication (≡⇒↝ ⌊ k ⌋-sym (sym eq)) ≡
to-implication (inverse (≡⇒↝ ⌊ k ⌋-sym eq))
≡⇒↝-sym k {A = A} {eq = eq} = elim¹
(λ eq → to-implication (≡⇒↝ ⌊ k ⌋-sym (sym eq)) ≡
to-implication (inverse (≡⇒↝ ⌊ k ⌋-sym eq)))
(to-implication (≡⇒↝ ⌊ k ⌋-sym (sym (refl A)))      ≡⟨ cong (to-implication ∘ ≡⇒↝ ⌊ k ⌋-sym) sym-refl ⟩
to-implication (≡⇒↝ ⌊ k ⌋-sym (refl A))            ≡⟨ cong (to-implication {k = ⌊ k ⌋-sym}) ≡⇒↝-refl ⟩
to-implication {k = ⌊ k ⌋-sym} id                  ≡⟨ to-implication-id ⌊ k ⌋-sym ⟩
id                                                 ≡⟨ sym \$ to-implication-inverse-id k ⟩
to-implication (inverse {k = k} id)                ≡⟨ cong (to-implication ∘ inverse {k = k}) \$ sym ≡⇒↝-refl ⟩∎
to-implication (inverse (≡⇒↝ ⌊ k ⌋-sym (refl A)))  ∎)
eq

≡⇒↝-trans : ∀ k {ℓ} {A B C : Set ℓ} {A≡B : A ≡ B} {B≡C : B ≡ C} →
to-implication (≡⇒↝ k (trans A≡B B≡C)) ≡
to-implication (≡⇒↝ k B≡C ∘ ≡⇒↝ k A≡B)
≡⇒↝-trans k {B = B} {A≡B = A≡B} = elim¹
(λ B≡C → to-implication (≡⇒↝ k (trans A≡B B≡C)) ≡
to-implication (≡⇒↝ k B≡C ∘ ≡⇒↝ k A≡B))
(to-implication (≡⇒↝ k (trans A≡B (refl B)))             ≡⟨ cong (to-implication ∘ ≡⇒↝ k) \$ trans-reflʳ _ ⟩
to-implication (≡⇒↝ k A≡B)                              ≡⟨ sym \$ cong (λ f → f ∘ to-implication (≡⇒↝ k A≡B)) \$ to-implication-id k ⟩
to-implication {k = k} id ∘ to-implication (≡⇒↝ k A≡B)  ≡⟨ sym \$ to-implication-∘ k ⟩
to-implication (id ∘ ≡⇒↝ k A≡B)                         ≡⟨ sym \$ cong (λ f → to-implication (f ∘ ≡⇒↝ k A≡B)) ≡⇒↝-refl ⟩∎
to-implication (≡⇒↝ k (refl B) ∘ ≡⇒↝ k A≡B)             ∎)
_

-- One can sometimes "push" ≡⇒↝ through cong.
--
-- This is a generalisation of a lemma due to Thierry Coquand.

≡⇒↝-cong : ∀ {k ℓ p A B} {eq : A ≡ B}
(P : Set ℓ → Set p)
(P-cong : ∀ {A B} → A ↝[ k ] B → P A ↝[ k ] P B) →
P-cong (id {A = A}) ≡ id →
≡⇒↝ _ (cong P eq) ≡ P-cong (≡⇒↝ _ eq)
≡⇒↝-cong {eq = eq} P P-cong P-cong-id = elim¹
(λ eq → ≡⇒↝ _ (cong P eq) ≡ P-cong (≡⇒↝ _ eq))
(≡⇒↝ _ (cong P (refl _))  ≡⟨ cong (≡⇒↝ _) \$ cong-refl P ⟩
≡⇒↝ _ (refl _)           ≡⟨ elim-refl (λ {A B} _ → A ↝[ _ ] B) _ ⟩
id                       ≡⟨ sym P-cong-id ⟩
P-cong id                ≡⟨ cong P-cong \$ sym \$
elim-refl (λ {A B} _ → A ↝[ _ ] B) _ ⟩∎
P-cong (≡⇒↝ _ (refl _))  ∎)
eq

-- One can express subst in terms of ≡⇒↝.

subst-in-terms-of-≡⇒↝ :
∀ k {a p} {A : Set a} {x y} (x≡y : x ≡ y) (P : A → Set p) p →
subst P x≡y p ≡ to-implication (≡⇒↝ k (cong P x≡y)) p
subst-in-terms-of-≡⇒↝ k x≡y P p = elim¹

(λ eq → subst P eq p ≡ to-implication (≡⇒↝ k (cong P eq)) p)

(subst P (refl _) p                          ≡⟨ subst-refl P p ⟩
p                                           ≡⟨ sym \$ cong (_\$ p) (to-implication-id k) ⟩
to-implication {k = k} id p                 ≡⟨ sym \$ cong (λ f → to-implication {k = k} f p) ≡⇒↝-refl ⟩
to-implication (≡⇒↝ k (refl _)) p           ≡⟨ sym \$ cong (λ eq → to-implication (≡⇒↝ k eq) p) \$ cong-refl P ⟩∎
to-implication (≡⇒↝ k (cong P (refl _))) p  ∎)

x≡y

subst-in-terms-of-inverse∘≡⇒↝ :
∀ k {a p} {A : Set a} {x y} (x≡y : x ≡ y) (P : A → Set p) p →
subst P (sym x≡y) p ≡
to-implication (inverse (≡⇒↝ ⌊ k ⌋-sym (cong P x≡y))) p
subst-in-terms-of-inverse∘≡⇒↝ k x≡y P p =
subst P (sym x≡y) p                                      ≡⟨ subst-in-terms-of-≡⇒↝ ⌊ k ⌋-sym (sym x≡y) P p ⟩
to-implication (≡⇒↝ ⌊ k ⌋-sym (cong P (sym x≡y))) p      ≡⟨ cong (λ eq → to-implication (≡⇒↝ ⌊ k ⌋-sym eq) p) (cong-sym P _) ⟩
to-implication (≡⇒↝ ⌊ k ⌋-sym (sym \$ cong P x≡y)) p      ≡⟨ cong (_\$ p) (≡⇒↝-sym k) ⟩∎
to-implication (inverse (≡⇒↝ ⌊ k ⌋-sym (cong P x≡y))) p  ∎

to-implication-≡⇒↝ :
∀ k {ℓ} {A B : Set ℓ} (eq : A ≡ B) →
to-implication (≡⇒↝ k eq) ≡ ≡⇒↝ implication eq
to-implication-≡⇒↝ k =
elim (λ eq → to-implication (≡⇒↝ k eq) ≡ ≡⇒↝ implication eq)
(λ A → to-implication (≡⇒↝ k (refl A))  ≡⟨ cong to-implication (≡⇒↝-refl {k = k}) ⟩
to-implication {k = k} id        ≡⟨ to-implication-id k ⟩
id                               ≡⟨ sym ≡⇒↝-refl ⟩∎
≡⇒↝ implication (refl A)         ∎)

------------------------------------------------------------------------
-- _⊎_ is a commutative monoid

-- _⊎_ preserves all kinds of functions.

private

⊎-cong-eq : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ⇔ A₂ → B₁ ⇔ B₂ → A₁ ⊎ B₁ ⇔ A₂ ⊎ B₂
⊎-cong-eq A₁⇔A₂ B₁⇔B₂ = record
{ to   = ⊎-map (to   A₁⇔A₂) (to   B₁⇔B₂)
; from = ⊎-map (from A₁⇔A₂) (from B₁⇔B₂)
} where open _⇔_

⊎-cong-inj : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↣ A₂ → B₁ ↣ B₂ → A₁ ⊎ B₁ ↣ A₂ ⊎ B₂
⊎-cong-inj A₁↣A₂ B₁↣B₂ = record
{ to        = to′
; injective = injective′
}
where
open _↣_

to′ = ⊎-map (to A₁↣A₂) (to B₁↣B₂)

abstract
injective′ : Injective to′
injective′ {x = inj₁ x} {y = inj₁ y} = cong inj₁ ⊚ injective A₁↣A₂ ⊚ ⊎.cancel-inj₁
injective′ {x = inj₂ x} {y = inj₂ y} = cong inj₂ ⊚ injective B₁↣B₂ ⊚ ⊎.cancel-inj₂
injective′ {x = inj₁ x} {y = inj₂ y} = ⊥-elim ⊚ ⊎.inj₁≢inj₂
injective′ {x = inj₂ x} {y = inj₁ y} = ⊥-elim ⊚ ⊎.inj₁≢inj₂ ⊚ sym

⊎-cong-surj : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↠ A₂ → B₁ ↠ B₂ → A₁ ⊎ B₁ ↠ A₂ ⊎ B₂
⊎-cong-surj A₁↠A₂ B₁↠B₂ = record
{ logical-equivalence = ⊎-cong-eq (_↠_.logical-equivalence A₁↠A₂)
(_↠_.logical-equivalence B₁↠B₂)
; right-inverse-of    =
[ cong inj₁ ⊚ _↠_.right-inverse-of A₁↠A₂
, cong inj₂ ⊚ _↠_.right-inverse-of B₁↠B₂
]
}

⊎-cong-bij : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↔ A₂ → B₁ ↔ B₂ → A₁ ⊎ B₁ ↔ A₂ ⊎ B₂
⊎-cong-bij A₁↔A₂ B₁↔B₂ = record
{ surjection      = ⊎-cong-surj (_↔_.surjection A₁↔A₂)
(_↔_.surjection B₁↔B₂)
; left-inverse-of =
[ cong inj₁ ⊚ _↔_.left-inverse-of A₁↔A₂
, cong inj₂ ⊚ _↔_.left-inverse-of B₁↔B₂
]
}

infixr 1 _⊎-cong_

_⊎-cong_ : ∀ {k a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↝[ k ] A₂ → B₁ ↝[ k ] B₂ → A₁ ⊎ B₁ ↝[ k ] A₂ ⊎ B₂
_⊎-cong_ {implication}         = ⊎-map
_⊎-cong_ {logical-equivalence} = ⊎-cong-eq
_⊎-cong_ {injection}           = ⊎-cong-inj
_⊎-cong_ {surjection}          = ⊎-cong-surj
_⊎-cong_ {bijection}           = ⊎-cong-bij
_⊎-cong_ {equivalence}         = λ A₁≃A₂ B₁≃B₂ →
from-bijection \$ ⊎-cong-bij (from-equivalence A₁≃A₂)
(from-equivalence B₁≃B₂)

-- _⊎_ is commutative.

⊎-comm : ∀ {a b} {A : Set a} {B : Set b} → A ⊎ B ↔ B ⊎ A
⊎-comm = record
{ surjection = record
{ logical-equivalence = record
{ to   = [ inj₂ , inj₁ ]
; from = [ inj₂ , inj₁ ]
}
; right-inverse-of = [ refl ⊚ inj₁ , refl ⊚ inj₂ ]
}
; left-inverse-of = [ refl ⊚ inj₁ , refl ⊚ inj₂ ]
}

-- _⊎_ is associative.

⊎-assoc : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
A ⊎ (B ⊎ C) ↔ (A ⊎ B) ⊎ C
⊎-assoc = record
{ surjection = record
{ logical-equivalence = record
{ to   = [ inj₁ ⊚ inj₁ , [ inj₁ ⊚ inj₂ , inj₂ ] ]
; from = [ [ inj₁ , inj₂ ⊚ inj₁ ] , inj₂ ⊚ inj₂ ]
}
; right-inverse-of =
[ [ refl ⊚ inj₁ ⊚ inj₁ , refl ⊚ inj₁ ⊚ inj₂ ] , refl ⊚ inj₂ ]
}
; left-inverse-of =
[ refl ⊚ inj₁ , [ refl ⊚ inj₂ ⊚ inj₁ , refl ⊚ inj₂ ⊚ inj₂ ] ]
}

-- ⊥ is a left and right identity of _⊎_.

⊎-left-identity : ∀ {a ℓ} {A : Set a} → ⊥ {ℓ = ℓ} ⊎ A ↔ A
⊎-left-identity = record
{ surjection = record
{ logical-equivalence = record
{ to   = λ { (inj₁ ()); (inj₂ x) → x }
; from = inj₂
}
; right-inverse-of = refl
}
; left-inverse-of = λ { (inj₁ ()); (inj₂ x) → refl (inj₂ x) }
}

⊎-right-identity : ∀ {a ℓ} {A : Set a} → A ⊎ ⊥ {ℓ = ℓ} ↔ A
⊎-right-identity {A = A} =
A ⊎ ⊥  ↔⟨ ⊎-comm ⟩
⊥ ⊎ A  ↔⟨ ⊎-left-identity ⟩□
A      □

-- For logical equivalences _⊎_ is also idempotent. (This lemma could
-- be generalised to cover surjections and implications.)

⊎-idempotent : ∀ {a} {A : Set a} → A ⊎ A ⇔ A
⊎-idempotent = record
{ to   = [ id , id ]
; from = inj₁
}

------------------------------------------------------------------------
-- _×_ is a commutative monoid with a zero

-- _×_ preserves all kinds of functions.

private

×-cong-eq : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ⇔ A₂ → B₁ ⇔ B₂ → A₁ × B₁ ⇔ A₂ × B₂
×-cong-eq A₁⇔A₂ B₁⇔B₂ = record
{ to   = Σ-map (to   A₁⇔A₂) (to   B₁⇔B₂)
; from = Σ-map (from A₁⇔A₂) (from B₁⇔B₂)
} where open _⇔_

×-cong-inj : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↣ A₂ → B₁ ↣ B₂ → A₁ × B₁ ↣ A₂ × B₂
×-cong-inj {A₁ = A₁} {A₂} {B₁} {B₂} A₁↣A₂ B₁↣B₂ = record
{ to        = to′
; injective = injective′
}
where
open _↣_

to′ : A₁ × B₁ → A₂ × B₂
to′ = Σ-map (to A₁↣A₂) (to B₁↣B₂)

abstract
injective′ : Injective to′
injective′ to′-x≡to′-y =
cong₂ _,_ (injective A₁↣A₂ (cong proj₁ to′-x≡to′-y))
(injective B₁↣B₂ (cong proj₂ to′-x≡to′-y))

×-cong-surj : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↠ A₂ → B₁ ↠ B₂ → A₁ × B₁ ↠ A₂ × B₂
×-cong-surj A₁↠A₂ B₁↠B₂ = record
{ logical-equivalence = ×-cong-eq (_↠_.logical-equivalence A₁↠A₂)
(_↠_.logical-equivalence B₁↠B₂)
; right-inverse-of    = uncurry λ x y →
cong₂ _,_ (_↠_.right-inverse-of A₁↠A₂ x)
(_↠_.right-inverse-of B₁↠B₂ y)
}

×-cong-bij : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↔ A₂ → B₁ ↔ B₂ → A₁ × B₁ ↔ A₂ × B₂
×-cong-bij A₁↔A₂ B₁↔B₂ = record
{ surjection      = ×-cong-surj (_↔_.surjection A₁↔A₂)
(_↔_.surjection B₁↔B₂)
; left-inverse-of = uncurry λ x y →
cong₂ _,_ (_↔_.left-inverse-of A₁↔A₂ x)
(_↔_.left-inverse-of B₁↔B₂ y)
}

infixr 2 _×-cong_

_×-cong_ : ∀ {k a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↝[ k ] A₂ → B₁ ↝[ k ] B₂ → A₁ × B₁ ↝[ k ] A₂ × B₂
_×-cong_ {implication}         = λ f g → Σ-map f g
_×-cong_ {logical-equivalence} = ×-cong-eq
_×-cong_ {injection}           = ×-cong-inj
_×-cong_ {surjection}          = ×-cong-surj
_×-cong_ {bijection}           = ×-cong-bij
_×-cong_ {equivalence}         = λ A₁≃A₂ B₁≃B₂ →
from-bijection \$ ×-cong-bij (from-equivalence A₁≃A₂)
(from-equivalence B₁≃B₂)

-- _×_ is commutative.

×-comm : ∀ {a b} {A : Set a} {B : Set b} → A × B ↔ B × A
×-comm = record
{ surjection = record
{ logical-equivalence = record
{ to   = uncurry λ x y → (y , x)
; from = uncurry λ x y → (y , x)
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}

-- Σ is associative.

Σ-assoc : ∀ {a b c}
{A : Set a} {B : A → Set b} {C : (x : A) → B x → Set c} →
(Σ A λ x → Σ (B x) (C x)) ↔ Σ (Σ A B) (uncurry C)
Σ-assoc = record
{ surjection = record
{ logical-equivalence = record
{ to   = λ { (x , (y , z)) → (x , y) , z }
; from = λ { ((x , y) , z) → x , (y , z) }
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}

-- _×_ is associative.

×-assoc : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
A × (B × C) ↔ (A × B) × C
×-assoc = Σ-assoc

-- ⊤ is a left and right identity of _×_ and Σ.

Σ-left-identity : ∀ {a} {A : ⊤ → Set a} → Σ ⊤ A ↔ A tt
Σ-left-identity = record
{ surjection = record
{ logical-equivalence = record
{ to   = proj₂
; from = λ x → (tt , x)
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}

×-left-identity : ∀ {a} {A : Set a} → ⊤ × A ↔ A
×-left-identity = Σ-left-identity

×-right-identity : ∀ {a} {A : Set a} → A × ⊤ ↔ A
×-right-identity {A = A} =
A × ⊤  ↔⟨ ×-comm ⟩
⊤ × A  ↔⟨ ×-left-identity ⟩□
A      □

-- ⊥ is a left and right zero of _×_ and Σ.

Σ-left-zero : ∀ {ℓ₁ a ℓ₂} {A : ⊥ {ℓ = ℓ₁} → Set a} →
Σ ⊥ A ↔ ⊥ {ℓ = ℓ₂}
Σ-left-zero = record
{ surjection = record
{ logical-equivalence = record
{ to   = λ { (() , _) }
; from = λ ()
}
; right-inverse-of = λ ()
}
; left-inverse-of = λ { (() , _) }
}

×-left-zero : ∀ {a ℓ₁ ℓ₂} {A : Set a} → ⊥ {ℓ = ℓ₁} × A ↔ ⊥ {ℓ = ℓ₂}
×-left-zero = Σ-left-zero

×-right-zero : ∀ {a ℓ₁ ℓ₂} {A : Set a} → A × ⊥ {ℓ = ℓ₁} ↔ ⊥ {ℓ = ℓ₂}
×-right-zero {A = A} =
A × ⊥  ↔⟨ ×-comm ⟩
⊥ × A  ↔⟨ ×-left-zero ⟩□
⊥      □

------------------------------------------------------------------------
-- Some lemmas related to Σ/∃/_×_

-- Σ preserves isomorphisms in its first argument and all kinds of
-- functions in its second argument.
--
-- The first two clauses are included as an optimisation intended to
-- make some proof terms easier to work with.

Σ-cong : ∀ {k₁ k₂ a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₁↔A₂ : A₁ ↔[ k₁ ] A₂) →
(∀ x → B₁ x ↝[ k₂ ] B₂ (to-implication A₁↔A₂ x)) →
Σ A₁ B₁ ↝[ k₂ ] Σ A₂ B₂
Σ-cong {equivalence} {equivalence} A₁≃A₂ B₁≃B₂ =
Eq.Σ-preserves A₁≃A₂ B₁≃B₂
Σ-cong {equivalence} {bijection} A₁≃A₂ B₁↔B₂ =
Eq.∃-preserves-bijections A₁≃A₂ B₁↔B₂
Σ-cong {k₁} {k₂} {A₁ = A₁} {A₂} {B₁} {B₂} A₁↔A₂ B₁↝B₂ = helper k₂ B₁↝B₂′
where
A₁≃A₂ : A₁ ≃ A₂
A₁≃A₂ = from-isomorphism A₁↔A₂

B₁↝B₂′ : ∀ x → B₁ x ↝[ k₂ ] B₂ (_≃_.to A₁≃A₂ x)
B₁↝B₂′ x =
B₁ x                                    ↝⟨ B₁↝B₂ x ⟩
B₂ (to-implication A₁↔A₂ x)             ↝⟨ ≡⇒↝ _ \$ cong (λ f → B₂ (f x)) \$
to-implication∘from-isomorphism k₁ equivalence ⟩
B₂ (_≃_.to (from-isomorphism A₁↔A₂) x)  □

helper : ∀ k₂ → (∀ x → B₁ x ↝[ k₂ ] B₂ (_≃_.to A₁≃A₂ x)) →
Σ A₁ B₁ ↝[ k₂ ] Σ A₂ B₂
helper implication         = Eq.∃-preserves-functions            A₁≃A₂
helper logical-equivalence = Eq.∃-preserves-logical-equivalences A₁≃A₂
helper injection           = Eq.∃-preserves-injections           A₁≃A₂
helper surjection          = Eq.∃-preserves-surjections          A₁≃A₂
helper bijection           = Eq.∃-preserves-bijections           A₁≃A₂
helper equivalence         = Eq.Σ-preserves                      A₁≃A₂

-- ∃ preserves all kinds of functions. One could define
-- ∃-cong = Σ-cong Bijection.id, but the resulting "from" functions
-- would contain an unnecessary use of substitutivity, and I want to
-- avoid that.

private

∃-cong-impl : ∀ {a b₁ b₂}
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x → B₂ x) → ∃ B₁ → ∃ B₂
∃-cong-impl B₁→B₂ = Σ-map id (λ {x} → B₁→B₂ x)

∃-cong-eq : ∀ {a b₁ b₂}
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ⇔ B₂ x) → ∃ B₁ ⇔ ∃ B₂
∃-cong-eq B₁⇔B₂ = record
{ to   = ∃-cong-impl (to   ⊚ B₁⇔B₂)
; from = ∃-cong-impl (from ⊚ B₁⇔B₂)
} where open _⇔_

∃-cong-surj : ∀ {a b₁ b₂}
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ↠ B₂ x) → ∃ B₁ ↠ ∃ B₂
∃-cong-surj B₁↠B₂ = record
{ logical-equivalence = ∃-cong-eq (_↠_.logical-equivalence ⊚ B₁↠B₂)
; right-inverse-of    = uncurry λ x y →
cong (_,_ x) (_↠_.right-inverse-of (B₁↠B₂ x) y)
}

∃-cong-bij : ∀ {a b₁ b₂}
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ↔ B₂ x) → ∃ B₁ ↔ ∃ B₂
∃-cong-bij B₁↔B₂ = record
{ surjection      = ∃-cong-surj (_↔_.surjection ⊚ B₁↔B₂)
; left-inverse-of = uncurry λ x y →
cong (_,_ x) (_↔_.left-inverse-of (B₁↔B₂ x) y)
}

∃-cong : ∀ {k a b₁ b₂}
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ↝[ k ] B₂ x) → ∃ B₁ ↝[ k ] ∃ B₂
∃-cong {implication}         = ∃-cong-impl
∃-cong {logical-equivalence} = ∃-cong-eq
∃-cong {injection}           = Σ-cong Bijection.id
∃-cong {surjection}          = ∃-cong-surj
∃-cong {bijection}           = ∃-cong-bij
∃-cong {equivalence}         = λ B₁≃B₂ →
from-bijection \$ ∃-cong-bij (from-equivalence ⊚ B₁≃B₂)

private

-- ∃-cong also works for _×_, in which case it is a more general
-- variant of id ×-cong_:

×-cong₂ : ∀ {k a b₁ b₂}
{A : Set a} {B₁ : Set b₁} {B₂ : Set b₂} →
(A → B₁ ↝[ k ] B₂) → A × B₁ ↝[ k ] A × B₂
×-cong₂ = ∃-cong

-- The following lemma is a more general variant of _×-cong id.

×-cong₁ : ∀ {k a₁ a₂ b}
{A₁ : Set a₁} {A₂ : Set a₂} {B : Set b} →
(B → A₁ ↝[ k ] A₂) → A₁ × B ↝[ k ] A₂ × B
×-cong₁ {A₁ = A₁} {A₂} {B} A₁↔A₂ =
A₁ × B  ↔⟨ ×-comm ⟩
B × A₁  ↝⟨ ∃-cong A₁↔A₂ ⟩
B × A₂  ↔⟨ ×-comm ⟩□
A₂ × B  □

-- Lemmas that can be used to simplify sigma types where one of the
-- two type arguments is (conditionally) isomorphic to the unit type.

drop-⊤-right : ∀ {k a b} {A : Set a} {B : A → Set b} →
((x : A) → B x ↔[ k ] ⊤) → Σ A B ↔ A
drop-⊤-right {A = A} {B} B↔⊤ =
Σ A B  ↔⟨ ∃-cong B↔⊤ ⟩
A × ⊤  ↝⟨ ×-right-identity ⟩□
A      □

drop-⊤-left-× : ∀ {k a b} {A : Set a} {B : Set b} →
(B → A ↔[ k ] ⊤) → A × B ↔ B
drop-⊤-left-× {A = A} {B} A↔⊤ =
A × B  ↝⟨ ×-comm ⟩
B × A  ↝⟨ drop-⊤-right A↔⊤ ⟩□
B      □

drop-⊤-left-Σ : ∀ {a b} {A : Set a} {B : A → Set b} →
(A↔⊤ : A ↔ ⊤) →
Σ A B ↔ B (_↔_.from A↔⊤ tt)
drop-⊤-left-Σ {A = A} {B} A↔⊤ =
Σ A B                   ↝⟨ inverse \$ Σ-cong (inverse A↔⊤) (λ _ → id) ⟩
Σ ⊤ (B ∘ _↔_.from A↔⊤)  ↝⟨ Σ-left-identity ⟩□
B (_↔_.from A↔⊤ tt)     □

-- Currying.

currying : ∀ {a b c} {A : Set a} {B : A → Set b} {C : Σ A B → Set c} →
((p : Σ A B) → C p) ↔ ((x : A) (y : B x) → C (x , y))
currying = record
{ surjection = record
{ logical-equivalence = record { to = curry; from = uncurry }
; right-inverse-of    = refl
}
; left-inverse-of = refl
}

-- Some lemmas relating functions from binary sums and pairs of
-- functions.

Π⊎↠Π×Π :
∀ {a b c} {A : Set a} {B : Set b} {C : A ⊎ B → Set c} →
((x : A ⊎ B) → C x)
↠
((x : A) → C (inj₁ x)) × ((y : B) → C (inj₂ y))
Π⊎↠Π×Π = record
{ logical-equivalence = record
{ to   = λ f → f ⊚ inj₁ , f ⊚ inj₂
; from = uncurry [_,_]
}
; right-inverse-of = refl
}

Π⊎↔Π×Π :
∀ {a b c} {A : Set a} {B : Set b} {C : A ⊎ B → Set c} →
Extensionality (a ⊔ b) c →
((x : A ⊎ B) → C x)
↔
((x : A) → C (inj₁ x)) × ((y : B) → C (inj₂ y))
Π⊎↔Π×Π ext = record
{ surjection      = Π⊎↠Π×Π
; left-inverse-of = λ _ → ext [ refl ⊚ _ , refl ⊚ _ ]
}

-- ∃ distributes "from the left" over _⊎_.

∃-⊎-distrib-left :
∀ {a b c} {A : Set a} {B : A → Set b} {C : A → Set c} →
(∃ λ x → B x ⊎ C x) ↔ ∃ B ⊎ ∃ C
∃-⊎-distrib-left = record
{ surjection = record
{ logical-equivalence = record
{ to   = uncurry λ x → [ inj₁ ⊚ _,_ x , inj₂ ⊚ _,_ x ]
; from = [ Σ-map id inj₁ , Σ-map id inj₂ ]
}
; right-inverse-of = [ refl ⊚ inj₁ , refl ⊚ inj₂ ]
}
; left-inverse-of =
uncurry λ x → [ refl ⊚ _,_ x ⊚ inj₁ , refl ⊚ _,_ x ⊚ inj₂ ]
}

-- ∃ also distributes "from the right" over _⊎_.

∃-⊎-distrib-right :
∀ {a b c} {A : Set a} {B : Set b} {C : A ⊎ B → Set c} →
Σ (A ⊎ B) C ↔ Σ A (C ⊚ inj₁) ⊎ Σ B (C ⊚ inj₂)
∃-⊎-distrib-right {A = A} {B} {C} = record
{ surjection = record
{ logical-equivalence = record
{ to   = to
; from = from
}
; right-inverse-of = [ refl ⊚ inj₁ , refl ⊚ inj₂ ]
}
; left-inverse-of = from∘to
}
where
to : Σ (A ⊎ B) C → Σ A (C ⊚ inj₁) ⊎ Σ B (C ⊚ inj₂)
to (inj₁ x , y) = inj₁ (x , y)
to (inj₂ x , y) = inj₂ (x , y)

from = [ Σ-map inj₁ id , Σ-map inj₂ id ]

from∘to : ∀ p → from (to p) ≡ p
from∘to (inj₁ x , y) = refl _
from∘to (inj₂ x , y) = refl _

-- ∃ is "commutative".

∃-comm : ∀ {a b c} {A : Set a} {B : Set b} {C : A → B → Set c} →
(∃ λ x → ∃ λ y → C x y) ↔ (∃ λ y → ∃ λ x → C x y)
∃-comm = record
{ surjection = record
{ logical-equivalence = record
{ to   = uncurry λ x → uncurry λ y z → (y , (x , z))
; from = uncurry λ x → uncurry λ y z → (y , (x , z))
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}

-- One can introduce an existential by also introducing an equality.

∃-intro : ∀ {a b} {A : Set a} (B : A → Set b) (x : A) →
B x ↔ ∃ λ y → B y × y ≡ x
∃-intro B x =
B x                    ↔⟨ inverse ×-right-identity ⟩
B x × ⊤                ↔⟨ id ×-cong _⇔_.to contractible⇔⊤↔ (singleton-contractible x) ⟩
B x × (∃ λ y → y ≡ x)  ↔⟨ ∃-comm ⟩
(∃ λ y → B x × y ≡ x)  ↔⟨ ∃-cong (λ y → ×-cong₁ (λ y≡x → subst (λ x → B x ↔ B y) y≡x id)) ⟩□
(∃ λ y → B y × y ≡ x)  □

-- A variant of ∃-intro.

∃-introduction :
∀ {a b} {A : Set a} {x : A} (B : (y : A) → x ≡ y → Set b) →
B x (refl x) ↔ ∃ λ y → ∃ λ (x≡y : x ≡ y) → B y x≡y
∃-introduction {x = x} B =
B x (refl x)                                              ↝⟨ ∃-intro (uncurry B) _ ⟩
(∃ λ { (y , x≡y) → B y x≡y × (y , x≡y) ≡ (x , refl x) })  ↝⟨ (∃-cong λ _ → ∃-cong λ _ →
inverse \$
_⇔_.to contractible⇔⊤↔ \$
mono₁ 0 (other-singleton-contractible x) _ _) ⟩
(∃ λ { (y , x≡y) → B y x≡y × ⊤ })                         ↝⟨ (∃-cong λ _ → ×-right-identity) ⟩
(∃ λ { (y , x≡y) → B y x≡y })                             ↝⟨ inverse Σ-assoc ⟩□
(∃ λ y → ∃ λ x≡y → B y x≡y)                               □

-- A non-dependent variant of Σ-≡,≡↔≡.

≡×≡↔≡ : ∀ {a b} {A : Set a} {B : Set b} {p₁ p₂ : A × B} →
(proj₁ p₁ ≡ proj₁ p₂ × proj₂ p₁ ≡ proj₂ p₂) ↔ (p₁ ≡ p₂)
≡×≡↔≡ {B = B} {p₁} {p₂} =
(proj₁ p₁ ≡ proj₁ p₂ × proj₂ p₁ ≡ proj₂ p₂)  ↝⟨ ∃-cong (λ _ → ≡⇒↝ _ \$ cong (λ q → q ≡ proj₂ p₂) \$
sym \$ subst-const _) ⟩
(∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) →
subst (λ _ → B) p (proj₂ p₁) ≡ proj₂ p₂)  ↝⟨ Bijection.Σ-≡,≡↔≡ ⟩□

(p₁ ≡ p₂)                                    □

-- If one is given an equality between pairs, where the second
-- components of the pairs are propositional, then one can restrict
-- attention to the first components.

ignore-propositional-component :
∀ {a b} {A : Set a} {B : A → Set b} {p q : Σ A B} →
Is-proposition (B (proj₁ q)) →
(proj₁ p ≡ proj₁ q) ↔ (p ≡ q)
ignore-propositional-component {B = B} {p₁ , p₂} {q₁ , q₂} Bq₁-prop =
(p₁ ≡ q₁)                                  ↝⟨ inverse ×-right-identity ⟩
(p₁ ≡ q₁ × ⊤)                              ↝⟨ ∃-cong (λ _ → _⇔_.to contractible⇔⊤↔ (Bq₁-prop _ _)) ⟩
(∃ λ (eq : p₁ ≡ q₁) → subst B eq p₂ ≡ q₂)  ↝⟨ Bijection.Σ-≡,≡↔≡ ⟩□
((p₁ , p₂) ≡ (q₁ , q₂))                    □

-- Contractible commutes with _×_ (assuming extensionality).

Contractible-commutes-with-× :
∀ {x y} {X : Set x} {Y : Set y} →
Extensionality (x ⊔ y) (x ⊔ y) →
Contractible (X × Y) ≃ (Contractible X × Contractible Y)
Contractible-commutes-with-× {x} {y} ext =
_↔_.to (Eq.⇔↔≃ ext
(Contractible-propositional ext)
(×-closure 1 (Contractible-propositional
(lower-extensionality y y ext))
(Contractible-propositional
(lower-extensionality x x ext))))
(record
{ to = λ cX×Y →
lemma cX×Y ,
lemma (H-level.respects-surjection
(_↔_.surjection ×-comm) 0 cX×Y)
; from = λ { ((x , eq₁) , (y , eq₂)) →
(x , y) ,
λ { (x′ , y′) →
(x  , y)   ≡⟨ cong₂ _,_ (eq₁ x′) (eq₂ y′) ⟩∎
(x′ , y′)  ∎ } }
})
where
lemma : ∀ {x y} {X : Set x} {Y : Set y} →
Contractible (X × Y) → Contractible X
lemma ((x , y) , eq) = x , λ x′ →
x               ≡⟨⟩
proj₁ (x , y)   ≡⟨ cong proj₁ (eq (x′ , y)) ⟩∎
proj₁ (x′ , y)  ∎

------------------------------------------------------------------------
-- Some lemmas related to _≃_

-- Equality of equivalences is isomorphic to pointwise equality of the
-- underlying functions (assuming extensionality).

≃-to-≡↔≡ :
∀ {a b} →
Extensionality (a ⊔ b) (a ⊔ b) →
{A : Set a} {B : Set b} {p q : A ≃ B} →
(∀ x → _≃_.to p x ≡ _≃_.to q x) ↔ p ≡ q
≃-to-≡↔≡ {a} {b} ext {p = p} {q} =
(∀ x → _≃_.to p x ≡ _≃_.to q x)                                        ↔⟨ Eq.extensionality-isomorphism (lower-extensionality b a ext) ⟩
_≃_.to p ≡ _≃_.to q                                                    ↝⟨ ignore-propositional-component (Eq.propositional ext _) ⟩
(_≃_.to p , _≃_.is-equivalence p) ≡ (_≃_.to q , _≃_.is-equivalence q)  ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ Eq.≃-as-Σ) ⟩□
p ≡ q                                                                  □

-- Equality of bijections between a set and a type is isomorphic to
-- pointwise equality of the underlying functions (assuming
-- extensionality).

↔-to-≡↔≡ :
∀ {a b} →
Extensionality (a ⊔ b) (a ⊔ b) →
{A : Set a} {B : Set b} {p q : A ↔ B} →
Is-set A →
(∀ x → _↔_.to p x ≡ _↔_.to q x) ↔ p ≡ q
↔-to-≡↔≡ ext {p = p} {q} A-set =
(∀ x → _↔_.to p x ≡ _↔_.to q x)                    ↝⟨ id ⟩
(∀ x → _≃_.to (Eq.↔⇒≃ p) x ≡ _≃_.to (Eq.↔⇒≃ q) x)  ↝⟨ ≃-to-≡↔≡ ext ⟩
Eq.↔⇒≃ p ≡ Eq.↔⇒≃ q                                ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ (Eq.↔↔≃ ext A-set)) ⟩□
p ≡ q                                              □

-- Equality of equivalences is isomorphic to pointwise equality of the
-- underlying /inverse/ functions (assuming extensionality).

≃-from-≡↔≡ :
∀ {a b} →
Extensionality (a ⊔ b) (a ⊔ b) →
{A : Set a} {B : Set b} {p q : A ≃ B} →
(∀ x → _≃_.from p x ≡ _≃_.from q x) ↔ p ≡ q
≃-from-≡↔≡ ext {p = p} {q} =
(∀ x → _≃_.from p x ≡ _≃_.from q x)  ↝⟨ ≃-to-≡↔≡ ext ⟩
inverse p ≡ inverse q                ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ (Eq.inverse-isomorphism ext)) ⟩□
p ≡ q                                □

-- Equality of bijections between a set and a type is isomorphic to
-- pointwise equality of the underlying /inverse/ functions (assuming
-- extensionality).

↔-from-≡↔≡ :
∀ {a b} →
Extensionality (a ⊔ b) (a ⊔ b) →
{A : Set a} {B : Set b} {p q : A ↔ B} →
Is-set A →
(∀ x → _↔_.from p x ≡ _↔_.from q x) ↔ p ≡ q
↔-from-≡↔≡ ext {p = p} {q} A-set =
(∀ x → _↔_.from p x ≡ _↔_.from q x)                    ↝⟨ id ⟩
(∀ x → _≃_.from (Eq.↔⇒≃ p) x ≡ _≃_.from (Eq.↔⇒≃ q) x)  ↝⟨ ≃-from-≡↔≡ ext ⟩
Eq.↔⇒≃ p ≡ Eq.↔⇒≃ q                                    ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ (Eq.↔↔≃ ext A-set)) ⟩□
p ≡ q                                                  □

-- Contractibility is isomorphic to equivalence to the unit type
-- (assuming extensionality).

contractible↔⊤≃ :
∀ {a} {A : Set a} →
Extensionality a a →
Contractible A ↔ (⊤ ≃ A)
contractible↔⊤≃ ext = record
{ surjection = record
{ logical-equivalence = record
{ to   = Eq.↔⇒≃ ∘ _⇔_.to contractible⇔⊤↔
; from = _⇔_.from contractible⇔⊤↔ ∘ _≃_.bijection
}
; right-inverse-of = λ _ →
Eq.lift-equality ext (refl _)
}
; left-inverse-of = λ _ →
_⇔_.to propositional⇔irrelevant
(Contractible-propositional ext) _ _
}

-- Equivalence to the empty type is equivalent to not being inhabited
-- (assuming extensionality).

≃⊥≃¬ :
∀ {a ℓ} {A : Set a} →
Extensionality (a ⊔ ℓ) (a ⊔ ℓ) →
(A ≃ ⊥ {ℓ = ℓ}) ≃ (¬ A)
≃⊥≃¬ {ℓ = ℓ} {A} ext =
_↔_.to (Eq.⇔↔≃ ext (Eq.right-closure ext 0 ⊥-propositional)
(¬-propositional
(lower-extensionality ℓ _ ext))) (record
{ to   = λ eq a → ⊥-elim (_≃_.to eq a)
; from = λ ¬a → A  ↔⟨ inverse (⊥↔uninhabited ¬a) ⟩□
⊥  □
})

------------------------------------------------------------------------
-- _⊎_ and _×_ form a commutative semiring

-- _×_ distributes from the left over _⊎_.

×-⊎-distrib-left : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
A × (B ⊎ C) ↔ (A × B) ⊎ (A × C)
×-⊎-distrib-left = ∃-⊎-distrib-left

-- _×_ distributes from the right over _⊎_.

×-⊎-distrib-right : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
(A ⊎ B) × C ↔ (A × C) ⊎ (B × C)
×-⊎-distrib-right = ∃-⊎-distrib-right

------------------------------------------------------------------------
-- Some lemmas related to functions

→-cong-⇔ : ∀ {a b c d}
{A : Set a} {B : Set b} {C : Set c} {D : Set d} →
A ⇔ B → C ⇔ D → (A → C) ⇔ (B → D)
→-cong-⇔ A⇔B C⇔D = record
{ to   = λ f → to   C⇔D ∘ f ∘ from A⇔B
; from = λ f → from C⇔D ∘ f ∘ to   A⇔B
}
where open _⇔_

→-cong : ∀ {a b c d} → Extensionality (a ⊔ b) (c ⊔ d) →
{A : Set a} {B : Set b} {C : Set c} {D : Set d} →
∀ {k} → A ↝[ ⌊ k ⌋-sym ] B → C ↝[ ⌊ k ⌋-sym ] D →
(A → C) ↝[ ⌊ k ⌋-sym ] (B → D)
→-cong {a} {b} {c} {d} ext {A} {B} {C} {D} = helper _
where
→-cong-↔ : A ↔ B → C ↔ D → (A → C) ↔ (B → D)
→-cong-↔ A↔B C↔D = record
{ surjection = record
{ logical-equivalence = logical-equiv
; right-inverse-of    = right-inv
}
; left-inverse-of = left-inv
}
where
open _↔_

logical-equiv = →-cong-⇔ (_↔_.logical-equivalence A↔B)
(_↔_.logical-equivalence C↔D)

abstract
right-inv :
∀ f → _⇔_.to logical-equiv (_⇔_.from logical-equiv f) ≡ f
right-inv f = lower-extensionality a c ext λ x →
to C↔D (from C↔D (f (to A↔B (from A↔B x))))  ≡⟨ right-inverse-of C↔D _ ⟩
f (to A↔B (from A↔B x))                      ≡⟨ cong f \$ right-inverse-of A↔B _ ⟩∎
f x                                          ∎

left-inv :
∀ f → _⇔_.from logical-equiv (_⇔_.to logical-equiv f) ≡ f
left-inv f = lower-extensionality b d ext λ x →
from C↔D (to C↔D (f (from A↔B (to A↔B x))))  ≡⟨ left-inverse-of C↔D _ ⟩
f (from A↔B (to A↔B x))                      ≡⟨ cong f \$ left-inverse-of A↔B _ ⟩∎
f x                                          ∎

helper : ∀ k → A ↝[ ⌊ k ⌋-sym ] B → C ↝[ ⌊ k ⌋-sym ] D →
(A → C) ↝[ ⌊ k ⌋-sym ] (B → D)
helper logical-equivalence      A⇔B C⇔D = →-cong-⇔ A⇔B C⇔D
helper bijection        A↔B C↔D = →-cong-↔ A↔B C↔D
helper equivalence A≃B C≃D = record
{ to             = to
; is-equivalence = λ y →
((from y , right-inverse-of y) , irrelevance y)
}
where
A→B≃C→D = Eq.↔⇒≃
(→-cong-↔ (_≃_.bijection A≃B) (_≃_.bijection C≃D))

to   = _≃_.to   A→B≃C→D
from = _≃_.from A→B≃C→D

abstract
right-inverse-of : ∀ x → to (from x) ≡ x
right-inverse-of = _≃_.right-inverse-of A→B≃C→D

irrelevance : ∀ y (p : to ⁻¹ y) →
(from y , right-inverse-of y) ≡ p
irrelevance = _≃_.irrelevance A→B≃C→D

Π-left-identity : ∀ {a} {A : ⊤ → Set a} → ((x : ⊤) → A x) ↔ A tt
Π-left-identity = record
{ surjection = record
{ logical-equivalence = record
{ to   = λ f → f tt
; from = λ x _ → x
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}

-- A lemma that can be used to simplify a pi type where the domain is
-- isomorphic to the unit type.

drop-⊤-left-Π : ∀ {a b} {A : Set a} {B : A → Set b} →
Extensionality a b →
(A↔⊤ : A ↔ ⊤) → ((x : A) → B x) ↔ B (_↔_.from A↔⊤ tt)
drop-⊤-left-Π {A = A} {B} ext A↔⊤ =
((x : A) → B x)                 ↔⟨ inverse \$ Eq.Π-preserves ext (inverse \$ from-isomorphism A↔⊤) (λ _ → id) ⟩
((x : ⊤) → B (_↔_.from A↔⊤ x))  ↝⟨ Π-left-identity ⟩□
B (_↔_.from A↔⊤ tt)             □

→-right-zero : ∀ {a} {A : Set a} → (A → ⊤) ↔ ⊤
→-right-zero = record
{ surjection = record
{ logical-equivalence = record
{ to   = λ _ → tt
; from = λ _ _ → tt
}
; right-inverse-of = λ _ → refl tt
}
; left-inverse-of = λ _ → refl (λ _ → tt)
}

-- Function types with the empty type as domain are isomorphic to the
-- unit type (assuming extensionality).

Π⊥↔⊤ : ∀ {ℓ a} {A : ⊥ {ℓ = ℓ} → Set a} →
Extensionality ℓ a →
((x : ⊥) → A x) ↔ ⊤
Π⊥↔⊤ ext = record
{ surjection = record
{ logical-equivalence = record
{ to   = _
; from = λ _ x → ⊥-elim x
}
; right-inverse-of = λ _ → refl _
}
; left-inverse-of = λ _ → ext (λ x → ⊥-elim x)
}

-- ¬ ⊥ is isomorphic to ⊤ (assuming extensionality).

¬⊥↔⊤ : ∀ {ℓ} →
Extensionality ℓ lzero →
¬ ⊥ {ℓ = ℓ} ↔ ⊤
¬⊥↔⊤ = Π⊥↔⊤

-- Simplification lemmas for types of the form A → A → B.

→→↠→ :
∀ {a b} {A : Set a} {B : Set b} →
(A → A → B) ↠ (A → B)
→→↠→ = record
{ logical-equivalence = record
{ to   = λ f x → f x x
; from = λ f x _ → f x
}
; right-inverse-of = refl
}

→→⊥↔→⊥ :
∀ {a ℓ} {A : Set a} →
Extensionality a (a ⊔ ℓ) →
(A → A → ⊥ {ℓ = ℓ}) ↔ (A → ⊥ {ℓ = ℓ})
→→⊥↔→⊥ ext = record
{ surjection      = →→↠→
; left-inverse-of = λ f → ext λ x → ⊥-elim (f x x)
}

-- Π is "commutative".

Π-comm : ∀ {a b c} {A : Set a} {B : Set b} {C : A → B → Set c} →
(∀ x y → C x y) ↔ (∀ y x → C x y)
Π-comm = record
{ surjection = record
{ logical-equivalence = record { to = flip; from = flip }
; right-inverse-of    = refl
}
; left-inverse-of = refl
}

-- Π and Σ commute (in a certain sense).

ΠΣ-comm :
∀ {a b c} {A : Set a} {B : A → Set b} {C : (x : A) → B x → Set c} →
(∀ x → ∃ λ (y : B x) → C x y)
↔
(∃ λ (f : ∀ x → B x) → ∀ x → C x (f x))
ΠΣ-comm = record
{ surjection = record
{ logical-equivalence = record
{ to   = λ f → proj₁ ⊚ f , proj₂ ⊚ f
; from = λ { (f , g) x → f x , g x }
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}

-- The Yoneda lemma, as given in the HoTT book, but specialised to the
-- opposite of the category of sets and functions, and with some
-- naturality properties omitted. (The proof uses extensionality.)

yoneda :
∀ {a b X} →
Extensionality (lsuc a) (lsuc a ⊔ b) →
(F : SET a → SET b) →
(map : ∀ {A B} → (Type A → Type B) → Type (F A) → Type (F B)) →
(∀ {A} {x : Type (F A)} → map id x ≡ x) →
(∀ {A B C f g x} →
(map {A = B} {B = C} f ∘ map {A = A} g) x ≡ map (f ∘ g) x) →

Type (F X)
↔
∃ λ (γ : ∀ Y → (Type X → Type Y) → Type (F Y)) →
∀ Y₁ Y₂ f g → map f (γ Y₁ g) ≡ γ Y₂ (f ∘ g)

yoneda {a} {X = X} ext F map map-id map-∘ = record
{ surjection = record
{ logical-equivalence = record
{ to = λ x → (λ _ f → map f x) , λ _ _ f g →
map f (map g x)  ≡⟨ map-∘ ⟩∎
map (f ∘ g) x    ∎
; from = λ { (γ , _) → γ X id }
}
; right-inverse-of = λ { (γ , h) → Σ-≡,≡→≡

((λ _ f → map f (γ X id))  ≡⟨ (lower-extensionality lzero (lsuc a) ext λ Y →
lower-extensionality _     (lsuc a) ext λ f →
h X Y f id) ⟩∎
(λ Y f → γ Y f)           ∎)

(_⇔_.to propositional⇔irrelevant
(Π-closure                                      ext  1 λ _  →
Π-closure (lower-extensionality lzero (lsuc a) ext) 1 λ Y₂ →
Π-closure (lower-extensionality _     (lsuc a) ext) 1 λ _  →
Π-closure (lower-extensionality _     (lsuc a) ext) 1 λ _  →
proj₂ (F Y₂) _ _)
_ _) }
}
; left-inverse-of = λ x →
map id x  ≡⟨ map-id ⟩∎
x         ∎
}

-- There is a (split) surjection from products of equality
-- isomorphisms to equalities.

Π≡↔≡-↠-≡ : ∀ k {a} {A : Set a} (x y : A) →
(∀ z → (z ≡ x) ↔[ k ] (z ≡ y)) ↠ (x ≡ y)
Π≡↔≡-↠-≡ k x y = record
{ logical-equivalence = record { to = to; from = from }
; right-inverse-of    = to∘from
}
where
to : (∀ z → (z ≡ x) ↔[ k ] (z ≡ y)) → x ≡ y
to f = to-implication (f x) (refl x)

from′ : x ≡ y → ∀ z → (z ≡ x) ↔ (z ≡ y)
from′ x≡y z = record
{ surjection = record
{ logical-equivalence = record
{ to   = λ z≡x → trans z≡x      x≡y
; from = λ z≡y → trans z≡y (sym x≡y)
}
; right-inverse-of = λ z≡y → trans-[trans-sym]- z≡y x≡y
}
; left-inverse-of = λ z≡x → trans-[trans]-sym z≡x x≡y
}

from : x ≡ y → ∀ z → (z ≡ x) ↔[ k ] (z ≡ y)
from x≡y z = from-bijection (from′ x≡y z)

abstract
to∘from : ∀ x≡y → to (from x≡y) ≡ x≡y
to∘from x≡y =
to (from x≡y)       ≡⟨ sym \$ cong (λ f → f (refl x)) \$ to-implication∘from-isomorphism bijection ⌊ k ⌋-iso ⟩
trans (refl x) x≡y  ≡⟨ trans-reflˡ _ ⟩∎
x≡y                 ∎

-- Products of equivalences of equalities are isomorphic to equalities
-- (assuming extensionality).

Π≡≃≡-↔-≡ : ∀ {a} → Extensionality a a →
{A : Set a} (x y : A) →
(∀ z → (z ≡ x) ≃ (z ≡ y)) ↔ (x ≡ y)
Π≡≃≡-↔-≡ ext x y = record
{ surjection      = surj
; left-inverse-of = from∘to
}
where
surj = Π≡↔≡-↠-≡ equivalence x y

open _↠_ surj

abstract
from∘to : ∀ f → from (to f) ≡ f
from∘to f = ext λ z → Eq.lift-equality ext \$ ext λ z≡x →
trans z≡x (_≃_.to (f x) (refl x))  ≡⟨ elim (λ {u v} u≡v →
(f : ∀ z → (z ≡ v) ≃ (z ≡ y)) →
trans u≡v (_≃_.to (f v) (refl v)) ≡
_≃_.to (f u) u≡v)
(λ _ _ → trans-reflˡ _)
z≡x f ⟩∎
_≃_.to (f z) z≡x                   ∎

-- One can introduce a universal quantifier by also introducing an
-- equality (assuming extensionality).

∀-intro : ∀ {a b} →
Extensionality a (a ⊔ b) →
{A : Set a} {x : A} (B : (y : A) → x ≡ y → Set b) →
B x (refl x) ↔ (∀ y (x≡y : x ≡ y) → B y x≡y)
∀-intro {a} ext {x = x} B = record
{ surjection = record
{ logical-equivalence = record
{ to   = to
; from = λ f → f x (refl x)
}
; right-inverse-of = to∘from
}
; left-inverse-of = from∘to
}
where
to : B x (refl x) → ∀ y (x≡y : x ≡ y) → B y x≡y
to b y x≡y =
subst (uncurry B)
(proj₂ (other-singleton-contractible x) (y , x≡y))
b

abstract

from∘to : ∀ b → to b x (refl x) ≡ b
from∘to b =
subst (uncurry B)
(proj₂ (other-singleton-contractible x) (x , refl x)) b  ≡⟨ cong (λ eq → subst (uncurry B) eq b) \$
other-singleton-contractible-refl x ⟩
subst (uncurry B) (refl (x , refl x)) b                        ≡⟨ subst-refl (uncurry B) _ ⟩∎
b                                                              ∎

to∘from : ∀ f → to (f x (refl x)) ≡ f
to∘from f = ext λ y → lower-extensionality lzero a ext λ x≡y →
elim¹ (λ {y} x≡y →
subst (uncurry B)
(proj₂ (other-singleton-contractible x) (y , x≡y))
(f x (refl x)) ≡
f y x≡y)
(subst (uncurry B)
(proj₂ (other-singleton-contractible x) (x , refl x))
(f x (refl x))                                         ≡⟨ from∘to (f x (refl x)) ⟩∎
f x (refl x)                                                 ∎)
x≡y

private

-- The following proof is perhaps easier to follow, but the
-- resulting "from" function is more complicated than the one used
-- in ∀-intro. (If subst reduced in the usual way when applied to
-- refl, then the two functions would be definitionally equal.)
--
-- This proof is based on one presented by Egbert Rijke in "A type
-- theoretical Yoneda lemma"
-- (http://homotopytypetheory.org/2012/05/02/a-type-theoretical-yoneda-lemma/).

∀-intro′ : ∀ {a b}```