------------------------------------------------------------------------
-- 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 {} {A B : Set }  A  B  A ↝[ k ] B
≡⇒↝ k = elim  {A B} _  A ↝[ k ] B)  _  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  {A B} _  A ↝[ k ] B) _

  ≡⇒↝-sym :  k {} {A B : Set } {eq : A  B} 
            to-implication (≡⇒↝  k ⌋-sym (sym eq)) 
            to-implication (inverse (≡⇒↝  k ⌋-sym eq))
  ≡⇒↝-sym k {A = A} {eq = eq} = elim¹
     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 {} {A B C : Set } {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
     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  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  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  p A B} {eq : A  B}
             (P : Set   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
     eq  ≡⇒↝ _ (cong P eq)  P-cong (≡⇒↝ _ eq))
    (≡⇒↝ _ (cong P (refl _))  ≡⟨ cong (≡⇒↝ _) $ cong-refl P 
     ≡⇒↝ _ (refl _)           ≡⟨ elim-refl  {A B} _  A ↝[ _ ] B) _ 
     id                       ≡⟨ sym P-cong-id 
     P-cong id                ≡⟨ cong P-cong $ sym $
                                   elim-refl  {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

     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  f  to-implication {k = k} f p) ≡⇒↝-refl 
     to-implication (≡⇒↝ k (refl _)) p           ≡⟨ sym $ cong  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  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 {} {A B : Set } (eq : A  B) 
    to-implication (≡⇒↝ k eq)  ≡⇒↝ implication eq
  to-implication-≡⇒↝ k =
    elim  eq  to-implication (≡⇒↝ k eq)  ≡⇒↝ implication eq)
          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 } {A : Set a}   { = }  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 } {A : Set a}  A   { = }  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 :  { = ℓ₁}  Set a} 
              Σ  A   { = 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}   { = ℓ₁} × A   { = ℓ₂}
×-left-zero = Σ-left-zero

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

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

-- See also Σ-left-zero and Σ-right-zero above.

-- 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  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  {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↔⊤)  _  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  y  ×-cong₁  y≡x  subst  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  _  ≡⇒↝ _ $ cong  q  q  proj₂ p₂) $
                                                                  sym $ subst-const _) 
  ( λ (p : proj₁ p₁  proj₁ p₂) 
     subst  _  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  _  _⇔_.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 } {A : Set a} 
  Extensionality (a  ) (a  ) 
  (A   { = })  (¬ A)
≃⊥≃¬ { = } {A} ext =
  _↔_.to (Eq.⇔↔≃ ext (Eq.right-closure ext 0 ⊥-propositional)
                     (¬-propositional
                        (lower-extensionality  _ 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↔⊤)  _  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  _  tt)
  }

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

Π⊥↔⊤ :  { a} {A :  { = }  Set a} 
       Extensionality  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  x  ⊥-elim x)
  }

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

¬⊥↔⊤ :  {} 
       Extensionality  lzero 
       ¬  { = }  
¬⊥↔⊤ = 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 } {A : Set a} 
  Extensionality a (a  ) 
  (A  A   { = })  (A   { = })
→→⊥↔→⊥ 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   _ 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) ⟩∎
          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  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  {u v} u≡v 
                                                    (f :  z  (z  v)  (z  y)) 
                                                    trans u≡v (_≃_.to (f v) (refl v)) 
                                                    _≃_.to (f u) u≡v)
                                                  _ _  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  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  {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)                                                 )