------------------------------------------------------------------------
-- Equivalences
------------------------------------------------------------------------

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

-- Partly based on Voevodsky's work on so-called univalent
-- foundations.

open import Equality

module Equivalence
  {reflexive} (eq :  {a p}  Equality-with-J a p reflexive) where

open import Bijection eq as Bijection hiding (id; _∘_; inverse)
open Derived-definitions-and-properties eq
open import Groupoid eq
open import H-level eq as H-level
open import H-level.Closure eq
open import Injection eq using (_↣_; module _↣_; Injective)
open import Logical-equivalence hiding (id; _∘_; inverse)
open import Preimage eq as Preimage using (_⁻¹_)
open import Prelude as P hiding (id) renaming (_∘_ to _⊚_)
open import Surjection eq as Surjection using (_↠_; module _↠_)

------------------------------------------------------------------------
-- Is-equivalence

-- A function f is an equivalence if all preimages under f are
-- contractible.

Is-equivalence :  {a b} {A : Set a} {B : Set b} 
                 (A  B)  Set (a  b)
Is-equivalence f =  y  Contractible (f ⁻¹ y)

abstract

  -- Is-equivalence f is a proposition, assuming extensional equality.

  propositional :
     {a b}  Extensionality (a  b) (a  b) 
    {A : Set a} {B : Set b} (f : A  B) 
    Is-proposition (Is-equivalence f)
  propositional {a} ext f =
    Π-closure (lower-extensionality a lzero ext) 1 λ _ 
      Contractible-propositional ext

-- Is-equivalence respects extensional equality.

respects-extensional-equality :
   {a b} {A : Set a} {B : Set b} {f g : A  B} 
  (∀ x  f x  g x) 
  Is-equivalence f  Is-equivalence g
respects-extensional-equality f≡g f-eq = λ b 
  H-level.respects-surjection
    (_↔_.surjection (Preimage.respects-extensional-equality f≡g))
    0
    (f-eq b)

abstract

  -- The function subst is an equivalence family.
  --
  -- Note that this proof would be easier if subst P (refl x) p
  -- reduced to p.

  subst-is-equivalence :
     {a p} {A : Set a} (P : A  Set p) {x y : A} (x≡y : x  y) 
    Is-equivalence (subst P x≡y)
  subst-is-equivalence P = elim
     {x y} x≡y  Is-equivalence (subst P x≡y))
     x p  _ , λ q 
       (p , subst-refl P p)                                     ≡⟨ elim
                                                                      {u v : P x} u≡v 
                                                                        _≡_ {A =  λ (w : P x)  subst P (refl x) w  v}
                                                                            (v , subst-refl P v)
                                                                            (u , trans (subst-refl P u) u≡v))
                                                                      p  cong (_,_ p) (sym $ trans-reflʳ _))
                                                                     (proj₁ q                     ≡⟨ sym $ subst-refl P (proj₁ q) 
                                                                      subst P (refl x) (proj₁ q)  ≡⟨ proj₂ q ⟩∎
                                                                      p                           ) 
       (proj₁ q , (trans      (subst-refl P (proj₁ q))  $
                   trans (sym (subst-refl P (proj₁ q))) $
                         proj₂ q))                              ≡⟨ cong (_,_ (proj₁ q)) $ sym $ trans-assoc _ _ _ 
       (proj₁ q , trans (trans      (subst-refl P (proj₁ q))
                               (sym (subst-refl P (proj₁ q))))
                        (proj₂ q))                              ≡⟨ cong  eq  proj₁ q , trans eq (proj₂ q)) $ trans-symʳ _ 
       (proj₁ q , trans (refl _) (proj₂ q))                     ≡⟨ cong (_,_ (proj₁ q)) $ trans-reflˡ _ ⟩∎
       q                                                        )

  -- If Σ-map id f is an equivalence, then f is also an equivalence.

  drop-Σ-map-id :
     {a b} {A : Set a} {B C : A  Set b} (f :  {x}  B x  C x) 
    Is-equivalence {A = Σ A B} {B = Σ A C} (Σ-map P.id f) 
     x  Is-equivalence (f {x = x})
  drop-Σ-map-id {b = b} {A} {B} {C} f eq x z =
    H-level.respects-surjection surj 0 (eq (x , z))
    where
    map-f : Σ A B  Σ A C
    map-f = Σ-map P.id f

    to-P :  {x y} {p :  C}  (x , f y)  p  Set b
    to-P {y = y} {p} _ =  λ y′  f y′  proj₂ p

    to : map-f ⁻¹ (x , z)  f ⁻¹ z
    to ((x′ , y) , eq) = elim¹ to-P (y , refl (f y)) eq

    from : f ⁻¹ z  map-f ⁻¹ (x , z)
    from (y , eq) = (x , y) , cong (_,_ x) eq

    to∘from :  p  to (from p)  p
    to∘from (y , eq) = elim¹
       {z′} (eq : f y  z′) 
         _≡_ {A =  λ (y : B x)  f y  z′}
             (elim¹ to-P (y , refl (f y)) (cong (_,_ x) eq))
             (y , eq))
      (elim¹ to-P (y , refl (f y)) (cong (_,_ x) (refl (f y)))  ≡⟨ cong (elim¹ to-P (y , refl (f y))) $
                                                                        cong-refl (_,_ x) {x = f y} 
       elim¹ to-P (y , refl (f y)) (refl (x , f y))             ≡⟨ elim¹-refl to-P _ ⟩∎
       (y , refl (f y))                                         )
      eq

    surj : map-f ⁻¹ (x , z)  f ⁻¹ z
    surj = record
      { logical-equivalence = record { to = to; from = from }
      ; right-inverse-of    = to∘from
      }

------------------------------------------------------------------------
-- _≃_

-- Equivalences.

infix 4 _≃_

record _≃_ {a b} (A : Set a) (B : Set b) : Set (a  b) where
  constructor ⟨_,_⟩
  field
    to             : A  B
    is-equivalence : Is-equivalence to

  -- Equivalent sets are isomorphic.

  from : B  A
  from y = proj₁ (proj₁ (is-equivalence y))

  right-inverse-of :  x  to (from x)  x
  right-inverse-of x = proj₂ (proj₁ (is-equivalence x))

  abstract

    left-inverse-of :  x  from (to x)  x
    left-inverse-of x =
      cong (proj₁ {B = λ x′  to x′  to x}) (
        proj₁ (is-equivalence (to x))  ≡⟨ proj₂ (is-equivalence (to x)) (x , refl (to x)) ⟩∎
        (x , refl (to x))              )

  bijection : A  B
  bijection = record
    { surjection = record
      { logical-equivalence = record
        { to   = to
        ; from = from
        }
      ; right-inverse-of = right-inverse-of
      }
    ; left-inverse-of = left-inverse-of
    }

  open _↔_ bijection public
    hiding (from; to; right-inverse-of; left-inverse-of)

  abstract

    -- All preimages of an element under the equivalence are equal.

    irrelevance :  y (p : to ⁻¹ y)  (from y , right-inverse-of y)  p
    irrelevance y = proj₂ (is-equivalence y)

    -- The two proofs left-inverse-of and right-inverse-of are
    -- related.

    left-right-lemma :
       x  cong to (left-inverse-of x)  right-inverse-of (to x)
    left-right-lemma x =
      lemma₁ to _ _ (lemma₂ (irrelevance (to x) (x , refl (to x))))
      where
      lemma₁ : {x y : A} (f : A  B) (p : x  y) (q : f x  f y) 
               refl (f y)  trans (cong f (sym p)) q 
               cong f p  q
      lemma₁ f = elim
         {x y} p   q  refl (f y)  trans (cong f (sym p)) q 
                           cong f p  q)
         x q hyp 
           cong f (refl x)                  ≡⟨ cong-refl f 
           refl (f x)                       ≡⟨ hyp 
           trans (cong f (sym (refl x))) q  ≡⟨ cong  p  trans (cong f p) q) sym-refl 
           trans (cong f (refl x)) q        ≡⟨ cong  p  trans p q) (cong-refl f) 
           trans (refl (f x)) q             ≡⟨ trans-reflˡ _ ⟩∎
           q                                )

      lemma₂ :  {f : A  B} {y} {f⁻¹y₁ f⁻¹y₂ : f ⁻¹ y}
               (p : f⁻¹y₁  f⁻¹y₂) 
               proj₂ f⁻¹y₂ 
               trans (cong f (sym (cong (proj₁ {B = λ x  f x  y}) p)))
                     (proj₂ f⁻¹y₁)
      lemma₂ {f} {y} =
        let pr = proj₁ {B = λ x  f x  y} in
        elim {A = f ⁻¹ y}
           {f⁻¹y₁ f⁻¹y₂} p 
             proj₂ f⁻¹y₂ 
               trans (cong f (sym (cong pr p))) (proj₂ f⁻¹y₁))
           f⁻¹y 
             proj₂ f⁻¹y                                               ≡⟨ sym $ trans-reflˡ _ 
             trans (refl (f (proj₁ f⁻¹y))) (proj₂ f⁻¹y)               ≡⟨ cong  p  trans p (proj₂ f⁻¹y)) (sym (cong-refl f)) 
             trans (cong f (refl (proj₁ f⁻¹y))) (proj₂ f⁻¹y)          ≡⟨ cong  p  trans (cong f p) (proj₂ f⁻¹y)) (sym sym-refl) 
             trans (cong f (sym (refl (proj₁ f⁻¹y)))) (proj₂ f⁻¹y)    ≡⟨ cong  p  trans (cong f (sym p)) (proj₂ f⁻¹y))
                                                                              (sym (cong-refl pr)) ⟩∎
             trans (cong f (sym (cong pr (refl f⁻¹y)))) (proj₂ f⁻¹y)  )

    right-left-lemma :
       x  cong from (right-inverse-of x)  left-inverse-of (from x)
    right-left-lemma x = subst
       x  cong from (right-inverse-of x)  left-inverse-of (from x))
      (right-inverse-of x)
      (let y = from x in

       cong from (right-inverse-of (to y))                          ≡⟨ cong (cong from) $ sym $ left-right-lemma y 
       cong from (cong to (left-inverse-of y))                      ≡⟨ cong-∘ from to _ 
       cong (from  to) (left-inverse-of y)                         ≡⟨ cong-roughly-id (from  to)  _  true) (left-inverse-of y)
                                                                                       _ _  z _  left-inverse-of z) 
       trans (left-inverse-of (from (to y)))
             (trans (left-inverse-of y) (sym (left-inverse-of y)))  ≡⟨ cong (trans _) $ trans-symʳ _ 
       trans (left-inverse-of (from (to y))) (refl _)               ≡⟨ trans-reflʳ _ ⟩∎
       left-inverse-of (from (to y))                                )

-- Bijections are equivalences.

↔⇒≃ :  {a b} {A : Set a} {B : Set b}  A  B  A  B
↔⇒≃ A↔B = record
  { to             = to
  ; is-equivalence = λ y 
      (from y , right-inverse-of y) , irrelevance y
  }
  where
  open _↔_ A↔B using (to; from)

  abstract
    is-equivalence : Is-equivalence to
    is-equivalence = Preimage.bijection⁻¹-contractible A↔B

    right-inverse-of :  x  to (from x)  x
    right-inverse-of = proj₂  proj₁  is-equivalence

    irrelevance :  y (p : to ⁻¹ y)  (from y , right-inverse-of y)  p
    irrelevance = proj₂  is-equivalence

------------------------------------------------------------------------
-- Equivalence

-- Equivalences are equivalence relations.

id :  {a} {A : Set a}  A  A
id =  P.id , singleton-contractible 

inverse :  {a b} {A : Set a} {B : Set b}  A  B  B  A
inverse A≃B =  from ,  y  (to y , left-inverse-of y) , irr y) 
  where
  open _≃_ A≃B

  abstract

    irr :  y (p : from ⁻¹ y)  (to y , left-inverse-of y)  p
    irr y (x , from-x≡y) =
      Σ-≡,≡→≡ (from-to from-x≡y) (elim¹
         {y} ≡y  subst  z  from z  y)
                          (trans (cong to (sym ≡y))
                                 (right-inverse-of x))
                          (left-inverse-of y)  ≡y)
        (let lemma =
               trans (cong to (sym (refl (from x))))
                     (right-inverse-of x)              ≡⟨ cong  eq  trans (cong to eq) (right-inverse-of x)) sym-refl 

               trans (cong to (refl (from x)))
                     (right-inverse-of x)              ≡⟨ cong  eq  trans eq (right-inverse-of x)) $ cong-refl to 

               trans (refl (to (from x)))
                     (right-inverse-of x)              ≡⟨ trans-reflˡ (right-inverse-of x) ⟩∎

               right-inverse-of x                      
         in

         subst  z  from z  from x)
               (trans (cong to (sym (refl (from x))))
                      (right-inverse-of x))
               (left-inverse-of (from x))                    ≡⟨ cong₂ (subst  z  from z  from x))
                                                                      lemma (sym $ right-left-lemma x) 
         subst  z  from z  from x)
               (right-inverse-of x)
               (cong from $ right-inverse-of x)              ≡⟨ subst-∘  z  z  from x) from _ 

         subst  z  z  from x)
               (cong from $ right-inverse-of x)
               (cong from $ right-inverse-of x)              ≡⟨ cong  eq  subst  z  z  from x) eq
                                                                                   (cong from $ right-inverse-of x)) $
                                                                     sym $ sym-sym _ 
         subst  z  z  from x)
               (sym $ sym $ cong from $ right-inverse-of x)
               (cong from $ right-inverse-of x)              ≡⟨ subst-trans _ 

         trans (sym $ cong from $ right-inverse-of x)
               (cong from $ right-inverse-of x)              ≡⟨ trans-symˡ _ ⟩∎

         refl (from x)                                       )
        from-x≡y)

infixr 9 _∘_

_∘_ :  {a b c} {A : Set a} {B : Set b} {C : Set c} 
      B  C  A  B  A  C
f  g = record
  { to             = to
  ; is-equivalence = λ y 
      (from y , right-inverse-of y) , irrelevance y
  }
  where
  f∘g  = ↔⇒≃ $ Bijection._∘_ (_≃_.bijection f) (_≃_.bijection g)
  to   = _≃_.to   f∘g
  from = _≃_.from f∘g

  abstract
    right-inverse-of :  x  to (from x)  x
    right-inverse-of = _≃_.right-inverse-of f∘g

    irrelevance :  y (p : to ⁻¹ y)  (from y , right-inverse-of y)  p
    irrelevance = _≃_.irrelevance f∘g

-- Equational reasoning combinators.

infix  -1 finally-≃
infixr -2 _≃⟨_⟩_

_≃⟨_⟩_ :  {a b c} (A : Set a) {B : Set b} {C : Set c} 
         A  B  B  C  A  C
_ ≃⟨ A≃B  B≃C = B≃C  A≃B

finally-≃ :  {a b} (A : Set a) (B : Set b)  A  B  A  B
finally-≃ _ _ A≃B = A≃B

syntax finally-≃ A B A≃B = A ≃⟨ A≃B ⟩□ B □

abstract

  -- Some simplification lemmas.

  right-inverse-of-id :
     {a} {A : Set a} {x : A} 
    _≃_.right-inverse-of id x  refl x
  right-inverse-of-id {x = x} = refl (refl x)

  left-inverse-of-id :
     {a} {A : Set a} {x : A} 
    _≃_.left-inverse-of id x  refl x
  left-inverse-of-id {x = x} =
     left-inverse-of x               ≡⟨⟩
     left-inverse-of (P.id x)        ≡⟨ sym $ right-left-lemma x 
     cong P.id (right-inverse-of x)  ≡⟨ sym $ cong-id _ 
     right-inverse-of x              ≡⟨ right-inverse-of-id ⟩∎
     refl x                          
     where open _≃_ id

  right-inverse-of∘inverse :
     {a b} {A : Set a} {B : Set b} 
     (A≃B : A  B) {x} 
    _≃_.right-inverse-of (inverse A≃B) x 
    _≃_.left-inverse-of A≃B x
  right-inverse-of∘inverse A≃B = refl _

  left-inverse-of∘inverse :
     {a b} {A : Set a} {B : Set b} 
     (A≃B : A  B) {x} 
    _≃_.left-inverse-of (inverse A≃B) x 
    _≃_.right-inverse-of A≃B x
  left-inverse-of∘inverse {A = A} {B} A≃B {x} =
    subst  x  _≃_.left-inverse-of (inverse A≃B) x 
                 right-inverse-of x)
          (right-inverse-of x)
          (_≃_.left-inverse-of (inverse A≃B) (to (from x))        ≡⟨ sym $ _≃_.right-left-lemma (inverse A≃B) (from x) 
           cong to (_≃_.right-inverse-of (inverse A≃B) (from x))  ≡⟨ cong (cong to) $ right-inverse-of∘inverse A≃B 
           cong to (left-inverse-of (from x))                     ≡⟨ left-right-lemma (from x) ⟩∎
           right-inverse-of (to (from x))                         )
    where open _≃_ A≃B

------------------------------------------------------------------------
-- The two-out-of-three property

-- If two out of three of f, g and g ∘ f are equivalences, then the
-- third one is also an equivalence.

record Two-out-of-three
         {a b c} {A : Set a} {B : Set b} {C : Set c}
         (f : A  B) (g : B  C) : Set (a  b  c) where
  field
    f-g   : Is-equivalence f  Is-equivalence g  Is-equivalence (g  f)
    g-g∘f : Is-equivalence g  Is-equivalence (g  f)  Is-equivalence f
    g∘f-f : Is-equivalence (g  f)  Is-equivalence f  Is-equivalence g

two-out-of-three :
   {a b c} {A : Set a} {B : Set b} {C : Set c}
  (f : A  B) (g : B  C)  Two-out-of-three f g
two-out-of-three f g = record
  { f-g   = λ f-eq g-eq 
              _≃_.is-equivalence ( g , g-eq    f , f-eq )
  ; g-g∘f = λ g-eq g∘f-eq 
              respects-extensional-equality
                 x  let g⁻¹ = _≃_.from  g , g-eq  in
                   g⁻¹ (g (f x))  ≡⟨ _≃_.left-inverse-of  g , g-eq  (f x) ⟩∎
                   f x            )
                (_≃_.is-equivalence
                   (inverse  g , g-eq    _ , g∘f-eq ))
  ; g∘f-f = λ g∘f-eq f-eq 
              respects-extensional-equality
                 x  let f⁻¹ = _≃_.from  f , f-eq  in
                   g (f (f⁻¹ x))  ≡⟨ cong g (_≃_.right-inverse-of  f , f-eq  x) ⟩∎
                   g x            )
                (_≃_.is-equivalence
                   ( _ , g∘f-eq   inverse  f , f-eq ))
  }

------------------------------------------------------------------------
-- f ≡ g and ∀ x → f x ≡ g x are isomorphic (assuming extensionality)

abstract

  -- Functions between contractible types are equivalences.

  function-between-contractible-types-is-equivalence :
     {a b} {A : Set a} {B : Set b} (f : A  B) 
    Contractible A  Contractible B  Is-equivalence f
  function-between-contractible-types-is-equivalence f cA cB =
    Two-out-of-three.g-g∘f
      (two-out-of-three f (const tt))
      (lemma cB)
      (lemma cA)
    where
    -- Functions from a contractible type to the unit type are
    -- contractible.

    lemma :  {b} {C : Set b}  Contractible C 
            Is-equivalence  (_ : C)  tt)
    lemma (x , irr) _ = (x , refl tt) , λ p 
      (x , refl tt)  ≡⟨ Σ-≡,≡→≡
                          (irr (proj₁ p))
                          (subst  _  tt  tt)
                             (irr (proj₁ p)) (refl tt)  ≡⟨ elim  eq  subst  _  tt  tt) eq (refl tt)  refl tt)
                                                                 _  subst-refl  _  tt  tt) (refl tt))
                                                                (irr (proj₁ p)) 
                           refl tt                      ≡⟨ elim  eq  refl tt  eq) (refl  refl) (proj₂ p) ⟩∎
                           proj₂ p                      ) ⟩∎
      p              

  -- ext⁻¹ is an equivalence (assuming extensionality).

  ext⁻¹-is-equivalence :
     {a b} {A : Set a} 
    ({B : A  Set b}  Extensionality′ A B) 
    {B : A  Set b} {f g : (x : A)  B x} 
    Is-equivalence (ext⁻¹ {f = f} {g = g})
  ext⁻¹-is-equivalence ext {f = f} {g} =
    let surj : (∀ x  Singleton (g x))  ( λ f   x  f x  g x)
        surj = record
          { logical-equivalence = record
            { to   = λ f  proj₁  f , proj₂  f
            ; from = λ p x  proj₁ p x , proj₂ p x
            }
          ; right-inverse-of = refl
          }

        lemma₁ : Contractible ( λ f   x  f x  g x)
        lemma₁ =
          H-level.respects-surjection surj 0 $
            _⇔_.from Π-closure-contractible⇔extensionality
              ext (singleton-contractible  g)

        lemma₂ : Is-equivalence (Σ-map P.id ext⁻¹)
        lemma₂ = function-between-contractible-types-is-equivalence
                   _ (singleton-contractible g) lemma₁

    in drop-Σ-map-id ext⁻¹ lemma₂ f

-- f ≡ g and ∀ x → f x ≡ g x are isomorphic (assuming extensionality).

extensionality-isomorphism :
   {a b} {A : Set a} 
  ({B : A  Set b}  Extensionality′ A B) 
  {B : A  Set b} {f g : (x : A)  B x} 
  (∀ x  f x  g x)  (f  g)
extensionality-isomorphism ext =
  inverse  _ , ext⁻¹-is-equivalence ext 

-- Note that the isomorphism gives us a really well-behaved notion of
-- extensionality.

good-ext :  {a b}  Extensionality a b  Extensionality a b
good-ext ext = _≃_.to (extensionality-isomorphism ext)

abstract

  good-ext-is-equivalence :
     {a b} (ext : Extensionality a b) 
    {A : Set a} {B : A  Set b} {f g : (x : A)  B x} 
    Is-equivalence {A =  x  f x  g x} (good-ext ext)
  good-ext-is-equivalence ext =
    _≃_.is-equivalence (extensionality-isomorphism ext)

  good-ext-refl :
     {a b} (ext : Extensionality a b)
    {A : Set a} {B : A  Set b} (f : (x : A)  B x) 
    good-ext ext  x  refl (f x))  refl f
  good-ext-refl ext f =
    _≃_.to (extensionality-isomorphism ext)  x  refl (f x))  ≡⟨ cong (_≃_.to (extensionality-isomorphism ext)) $ sym $
                                                                        ext  _  ext⁻¹-refl f) 
    _≃_.to (extensionality-isomorphism ext) (ext⁻¹ (refl f))    ≡⟨ _≃_.right-inverse-of (extensionality-isomorphism ext) _ ⟩∎
    refl f                                                      

  cong-good-ext :
     {a b} (ext : Extensionality a b)
    {A : Set a} {B : A  Set b} {f g : (x : A)  B x}
    (f≡g :  x  f x  g x) {x} 
    cong  h  h x) (good-ext ext f≡g)  f≡g x
  cong-good-ext ext f≡g {x} =
    let lemma = elim
           f≡g  cong  h  h x) f≡g  ext⁻¹ f≡g x)
           f  cong  h  h x) (refl f)  ≡⟨ cong-refl  h  h x) {x = f} 
                 refl (f x)                 ≡⟨ sym $ ext⁻¹-refl f ⟩∎
                 ext⁻¹ (refl f) x           )
          (good-ext ext f≡g)
    in

    cong  h  h x) (good-ext ext f≡g)  ≡⟨ lemma 
    ext⁻¹ (good-ext ext f≡g) x           ≡⟨ cong  h  h x) $
                                                 _≃_.left-inverse-of (extensionality-isomorphism ext) f≡g ⟩∎
    f≡g x                                

------------------------------------------------------------------------
-- Groupoid

abstract

  -- Two proofs of equivalence are equal if the function components
  -- are equal (assuming extensionality).

  lift-equality :
     {a b}  Extensionality (a  b) (a  b) 
    {A : Set a} {B : Set b} {p q : A  B} 
    _≃_.to p  _≃_.to q  p  q
  lift-equality {a} {b} ext {p =  f , f-eq } {q =  g , g-eq } f≡g =
    elim  {f g} f≡g   f-eq g-eq   f , f-eq    g , g-eq )
          f f-eq g-eq 
            cong (⟨_,_⟩ f)
              (_⇔_.to {To = Proof-irrelevant _}
                      propositional⇔irrelevant
                      (propositional ext f) f-eq g-eq))
         f≡g f-eq g-eq

  -- Two proofs of equivalence are equal if the /inverses/ of the
  -- function components are equal (assuming extensionality).

  lift-equality-inverse :
     {a b}  Extensionality (a  b) (a  b) 
    {A : Set a} {B : Set b} {p q : A  B} 
    _≃_.from p  _≃_.from q  p  q
  lift-equality-inverse ext {p = p} {q = q} f≡g =
    p                    ≡⟨ lift-equality ext (refl _) 
    inverse (inverse p)  ≡⟨ cong inverse $ lift-equality ext f≡g 
    inverse (inverse q)  ≡⟨ lift-equality ext (refl _) ⟩∎
    q                    

-- _≃_ comes with a groupoid structure (assuming extensionality).

groupoid :  {}  Extensionality    Groupoid (lsuc ) 
groupoid {} ext = record
  { Object         = Set 
  ; _∼_            = _≃_
  ; id             = id
  ; _∘_            = _∘_
  ; _⁻¹            = inverse
  ; left-identity  = left-identity
  ; right-identity = right-identity
  ; assoc          = assoc
  ; left-inverse   = left-inverse
  ; right-inverse  = right-inverse
  }
  where
  abstract
    left-identity : {X Y : Set } (p : X  Y)  id  p  p
    left-identity _ = lift-equality ext (refl _)

    right-identity : {X Y : Set } (p : X  Y)  p  id  p
    right-identity _ = lift-equality ext (refl _)

    assoc : {W X Y Z : Set } (p : Y  Z) (q : X  Y) (r : W  X) 
            p  (q  r)  (p  q)  r
    assoc _ _ _ = lift-equality ext (refl _)

    left-inverse : {X Y : Set } (p : X  Y)  inverse p  p  id
    left-inverse p = lift-equality ext (ext $ _≃_.left-inverse-of p)

    right-inverse : {X Y : Set } (p : X  Y)  p  inverse p  id
    right-inverse p = lift-equality ext (ext $ _≃_.right-inverse-of p)

------------------------------------------------------------------------
-- A surjection from A ↔ B to A ≃ B, and related results

private
 abstract

  -- ↔⇒≃ is a left inverse of _≃_.bijection (assuming extensionality).

  ↔⇒≃-left-inverse :
     {a b} {A : Set a} {B : Set b} 
    Extensionality (a  b) (a  b) 
    (A≃B : A  B) 
    ↔⇒≃ (_≃_.bijection A≃B)  A≃B
  ↔⇒≃-left-inverse ext _ = lift-equality ext (refl _)

  -- When sets are used ↔⇒≃ is a right inverse of _≃_.bijection
  -- (assuming extensionality).

  ↔⇒≃-right-inverse :
     {a b} {A : Set a} {B : Set b} 
    Extensionality (a  b) (a  b) 
    Is-set A  (A↔B : A  B) 
    _≃_.bijection (↔⇒≃ A↔B)  A↔B
  ↔⇒≃-right-inverse {a} {b} {B = B} ext A-set A↔B =
    cong₂  l r  record
             { surjection = record
               { logical-equivalence = _↔_.logical-equivalence A↔B
               ; right-inverse-of    = r
               }
             ; left-inverse-of = l
             })
          (lower-extensionality b b ext λ _  _⇔_.to set⇔UIP A-set _ _)
          (lower-extensionality a a ext λ _  _⇔_.to set⇔UIP B-set _ _)
    where
    B-set : Is-set B
    B-set = respects-surjection (_↔_.surjection A↔B) 2 A-set

-- There is a surjection from A ↔ B to A ≃ B (assuming
-- extensionality).

↔↠≃ :
   {a b} {A : Set a} {B : Set b} 
  Extensionality (a  b) (a  b) 
  (A  B)  (A  B)
↔↠≃ ext = record
  { logical-equivalence = record
    { to   = ↔⇒≃
    ; from = _≃_.bijection
    }
  ; right-inverse-of = ↔⇒≃-left-inverse ext
  }

-- When A is a set A ↔ B and A ≃ B are isomorphic (assuming
-- extensionality).

↔↔≃ :
   {a b} {A : Set a} {B : Set b} 
  Extensionality (a  b) (a  b) 
  Is-set A  (A  B)  (A  B)
↔↔≃ ext A-set = record
  { surjection      = ↔↠≃ ext
  ; left-inverse-of = ↔⇒≃-right-inverse ext A-set
  }

-- For propositional types logical equivalence is isomorphic to
-- equivalence (assuming extensionality).

⇔↔≃ :  {a b}  Extensionality (a  b) (a  b) 
      {A : Set a} {B : Set b} 
      Is-proposition A  Is-proposition B  (A  B)  (A  B)
⇔↔≃ ext {A} {B} A-prop B-prop = record
  { surjection = record
    { logical-equivalence = record
      { to   = ⇔→≃
      ; from = _≃_.logical-equivalence
      }
    ; right-inverse-of = λ _  lift-equality ext (refl _)
    }
  ; left-inverse-of = refl
  }
  where
  ⇔→≃ : A  B  A  B
  ⇔→≃ A⇔B = ↔⇒≃ record
    { surjection = record
      { logical-equivalence = A⇔B
      ; right-inverse-of    = to∘from
      }
    ; left-inverse-of = from∘to
    }
    where
    open _⇔_ A⇔B

    abstract

      to∘from :  x  to (from x)  x
      to∘from _ = _⇔_.to propositional⇔irrelevant B-prop _ _

      from∘to :  x  from (to x)  x
      from∘to _ = _⇔_.to propositional⇔irrelevant A-prop _ _

------------------------------------------------------------------------
-- Closure, preservation

abstract

  -- Positive h-levels are closed under the equivalence operator
  -- (assuming extensionality).

  right-closure :
     {a b}  Extensionality (a  b) (a  b) 
     {A : Set a} {B : Set b} n 
    H-level (1 + n) B  H-level (1 + n) (A  B)
  right-closure {a} {b} ext {A = A} {B} n h =
    H-level.respects-surjection surj (1 + n) lemma
    where
    lemma : H-level (1 + n) ( λ (to : A  B)  Is-equivalence to)
    lemma = Σ-closure (1 + n)
              (Π-closure (lower-extensionality b a ext)
                         (1 + n) (const h))
              (mono (m≤m+n 1 n)  propositional ext)

    surj : ( λ (to : A  B)  Is-equivalence to)  (A  B)
    surj = record
      { logical-equivalence = record
          { to   = λ A≃B   proj₁ A≃B , proj₂ A≃B 
          ; from = λ A≃B  (_≃_.to A≃B , _≃_.is-equivalence A≃B)
          }
      ; right-inverse-of = λ _  refl _
      }

  left-closure :
     {a b}  Extensionality (a  b) (a  b) 
     {A : Set a} {B : Set b} n 
    H-level (1 + n) A  H-level (1 + n) (A  B)
  left-closure ext {A = A} {B} n h =
    H-level.[inhabited⇒+]⇒+ n λ (A≃B : A  B) 
      right-closure ext n $
        H-level.respects-surjection (_≃_.surjection A≃B) (1 + n) h

-- Equalities are closed, in a strong sense, under applications of
-- equivalences.

≃-≡ :  {a b} {A : Set a} {B : Set b} (A≃B : A  B) {x y : A} 
      let open _≃_ A≃B in
      (to x  to y)  (x  y)
≃-≡ A≃B {x} {y} = ↔⇒≃ record
  { surjection      = surjection′
  ; left-inverse-of = left-inverse-of′
  }
  where
  open _≃_ A≃B

  surjection′ : (to x  to y)  (x  y)
  surjection′ =
    Surjection.↠-≡ $
    _↔_.surjection $
    Bijection.inverse $
    _≃_.bijection A≃B

  abstract
    left-inverse-of′ :
       p  _↠_.from surjection′ (_↠_.to surjection′ p)  p
    left-inverse-of′ = λ to-x≡to-y 
      cong to (
        trans (sym (left-inverse-of x)) $
        trans (cong from to-x≡to-y) $
        left-inverse-of y)                         ≡⟨ cong-trans to _ _ 
      trans (cong to (sym (left-inverse-of x))) (
        cong to (trans (cong from to-x≡to-y) (
                 left-inverse-of y)))              ≡⟨ cong₂ trans (cong-sym to _) (cong-trans to _ _) 
      trans (sym (cong to (left-inverse-of x))) (
        trans (cong to (cong from to-x≡to-y)) (
        cong to (left-inverse-of y)))              ≡⟨ cong₂  eq₁ eq₂  trans (sym eq₁) $
                                                                         trans (cong to (cong from to-x≡to-y)) $
                                                                         eq₂)
                                                           (left-right-lemma x)
                                                           (left-right-lemma y) 
      trans (sym (right-inverse-of (to x))) (
        trans (cong to (cong from to-x≡to-y)) (
        right-inverse-of (to y)))                  ≡⟨ _↠_.right-inverse-of (Surjection.↠-≡ $ _≃_.surjection A≃B) to-x≡to-y ⟩∎
      to-x≡to-y                                    

abstract
  private

    -- We can push subst through certain function applications.

    push-subst :
       {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
        (B₁ : A₁  Set b₁) {B₂ : A₂  Set b₂}
        {f : A₂  A₁} {x₁ x₂ : A₂} {y : B₁ (f x₁)}
      (g :  x  B₁ (f x)  B₂ x) (eq : x₁  x₂) 
      subst B₂ eq (g x₁ y)  g x₂ (subst B₁ (cong f eq) y)
    push-subst B₁ {B₂} {f} g eq = elim
       {x₁ x₂} eq   y  subst B₂ eq (g x₁ y) 
                            g x₂ (subst B₁ (cong f eq) y))
       x y  subst B₂ (refl x) (g x y)           ≡⟨ subst-refl B₂ _ 
               g x y                               ≡⟨ sym $ cong (g x) $ subst-refl B₁ _ 
               g x (subst B₁ (refl (f x)) y)       ≡⟨ cong  eq  g x (subst B₁ eq y)) (sym $ cong-refl f) ⟩∎
               g x (subst B₁ (cong f (refl x)) y)  )
      eq _

    push-subst′ :
       {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
      (A₁≃A₂ : A₁  A₂) (B₁ : A₁  Set b₁) (B₂ : A₂  Set b₂) 
      let open _≃_ A₁≃A₂ in {x₁ x₂ : A₁} {y : B₁ (from (to x₁))}
      (g :  x  B₁ (from (to x))  B₂ (to x)) (eq : to x₁  to x₂) 
      subst B₂ eq (g x₁ y)  g x₂ (subst B₁ (cong from eq) y)
    push-subst′ A₁≃A₂ B₁ B₂ {x₁} {x₂} {y} g eq =
      subst B₂ eq (g x₁ y)                    ≡⟨ cong (subst B₂ eq) $ sym $ g′-lemma _ _ 
      subst B₂ eq (g′ (to x₁) y)              ≡⟨ push-subst B₁ g′ eq 
      g′ (to x₂) (subst B₁ (cong from eq) y)  ≡⟨ g′-lemma _ _ ⟩∎
      g x₂ (subst B₁ (cong from eq) y)        
      where
      open _≃_ A₁≃A₂

      g′ :  x′  B₁ (from x′)  B₂ x′
      g′ x′ y = subst B₂ (right-inverse-of x′) $
                g (from x′) $
                subst B₁ (sym $ cong from $ right-inverse-of x′) y

      g′-lemma :  x y  g′ (to x) y  g x y
      g′-lemma x y =
        let lemma = λ y 
              let gy = g (from (to x)) $
                         subst B₁
                           (sym $ cong from $ cong to (refl _)) y in
              subst B₂ (cong to (refl _)) gy             ≡⟨ cong  p  subst B₂ p gy) $ cong-refl to 
              subst B₂ (refl _) gy                       ≡⟨ subst-refl B₂ gy 
              gy                                         ≡⟨ cong  p  g (from (to x)) $ subst B₁ (sym $ cong from p) y) $ cong-refl to 
              g (from (to x))
                (subst B₁ (sym $ cong from (refl _)) y)  ≡⟨ cong  p  g (from (to x)) $ subst B₁ (sym p) y) $ cong-refl from 
              g (from (to x))
                (subst B₁ (sym (refl _)) y)              ≡⟨ cong  p  g (from (to x)) $ subst B₁ p y) sym-refl 
              g (from (to x)) (subst B₁ (refl _) y)      ≡⟨ cong (g (from (to x))) $ subst-refl B₁ y ⟩∎
              g (from (to x)) y                          
        in
        subst B₂ (right-inverse-of (to x))
          (g (from (to x)) $
           subst B₁ (sym $ cong from $
                       right-inverse-of (to x)) y)  ≡⟨ cong  p  subst B₂ p (g (from (to x)) $ subst B₁ (sym $ cong from p) y)) $
                                                         sym $ left-right-lemma x 
        subst B₂ (cong to $ left-inverse-of x)
          (g (from (to x)) $
           subst B₁ (sym $ cong from $ cong to $
                       left-inverse-of x) y)        ≡⟨ elim¹
                                                          {x′} eq 
                                                            (y : B₁ (from (to x′))) 
                                                            subst B₂ (cong to eq)
                                                              (g (from (to x)) $ subst B₁ (sym $ cong from $ cong to eq) y) 
                                                            g x′ y)
                                                         lemma
                                                         (left-inverse-of x) y ⟩∎
        g x y                                       

-- If the first component is instantiated to id, then the following
-- lemmas state that ∃ preserves functions, logical equivalences,
-- injections, surjections and bijections.

∃-preserves-functions :
   {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
    {B₁ : A₁  Set b₁} {B₂ : A₂  Set b₂}
  (A₁≃A₂ : A₁  A₂)  (∀ x  B₁ x  B₂ (_≃_.to A₁≃A₂ x)) 
  Σ A₁ B₁  Σ A₂ B₂
∃-preserves-functions A₁≃A₂ B₁→B₂ = Σ-map (_≃_.to A₁≃A₂) (B₁→B₂ _)

∃-preserves-logical-equivalences :
   {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
    {B₁ : A₁  Set b₁} {B₂ : A₂  Set b₂}
  (A₁≃A₂ : A₁  A₂)  (∀ x  B₁ x  B₂ (_≃_.to A₁≃A₂ x)) 
  Σ A₁ B₁  Σ A₂ B₂
∃-preserves-logical-equivalences {B₂ = B₂} A₁≃A₂ B₁⇔B₂ = record
  { to   = ∃-preserves-functions A₁≃A₂ (_⇔_.to  B₁⇔B₂)
  ; from =
     ∃-preserves-functions
       (inverse A₁≃A₂)
        x y  _⇔_.from
                  (B₁⇔B₂ (_≃_.from A₁≃A₂ x))
                  (subst B₂ (sym (_≃_.right-inverse-of A₁≃A₂ x)) y))
  }

∃-preserves-injections :
   {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
    {B₁ : A₁  Set b₁} {B₂ : A₂  Set b₂}
  (A₁≃A₂ : A₁  A₂)  (∀ x  B₁ x  B₂ (_≃_.to A₁≃A₂ x)) 
  Σ A₁ B₁  Σ A₂ B₂
∃-preserves-injections {A₁ = A₁} {A₂} {B₁} {B₂} A₁≃A₂ B₁↣B₂ = record
  { to        = to′
  ; injective = injective′
  }
  where
  open _↣_

  to′ : Σ A₁ B₁  Σ A₂ B₂
  to′ = ∃-preserves-functions A₁≃A₂ (_↣_.to  B₁↣B₂)

  abstract
    injective′ : Injective to′
    injective′ {x = (x₁ , x₂)} {y = (y₁ , y₂)} =
      _↔_.to Σ-≡,≡↔≡ 
      Σ-map (_≃_.injective A₁≃A₂)  {eq₁} eq₂ 
        let lemma =
              to (B₁↣B₂ y₁) (subst B₁ (_≃_.injective A₁≃A₂ eq₁) x₂)      ≡⟨ refl _ 
              to (B₁↣B₂ y₁)
                (subst B₁ (trans (sym (_≃_.left-inverse-of A₁≃A₂ x₁)) $
                           trans (cong (_≃_.from A₁≃A₂) eq₁)
                                 (_≃_.left-inverse-of A₁≃A₂ y₁))
                          x₂)                                            ≡⟨ cong (to (B₁↣B₂ y₁)) $ sym $ subst-subst B₁ _ _ _ 
              to (B₁↣B₂ y₁)
                 (subst B₁ (trans (cong (_≃_.from A₁≃A₂) eq₁)
                                  (_≃_.left-inverse-of A₁≃A₂ y₁)) $
                  subst B₁ (sym (_≃_.left-inverse-of A₁≃A₂ x₁))
                           x₂)                                           ≡⟨ cong (to (B₁↣B₂ y₁)) $ sym $ subst-subst B₁ _ _ _ 
              to (B₁↣B₂ y₁)
                 (subst B₁ (_≃_.left-inverse-of A₁≃A₂ y₁) $
                  subst B₁ (cong (_≃_.from A₁≃A₂) eq₁) $
                  subst B₁ (sym (_≃_.left-inverse-of A₁≃A₂ x₁)) x₂)      ≡⟨ sym $ push-subst′ A₁≃A₂ B₁ B₂
                                                                               x y  to (B₁↣B₂ x)
                                                                                          (subst B₁ (_≃_.left-inverse-of A₁≃A₂ x) y))
                                                                              eq₁ 
              subst B₂ eq₁
                (to (B₁↣B₂ x₁) $
                 subst B₁ (_≃_.left-inverse-of A₁≃A₂ x₁) $
                 subst B₁ (sym (_≃_.left-inverse-of A₁≃A₂ x₁)) x₂)       ≡⟨ cong (subst B₂ eq₁  to (B₁↣B₂ x₁)) $ subst-subst B₁ _ _ _ 
              subst B₂ eq₁
                (to (B₁↣B₂ x₁) $
                 subst B₁ (trans (sym (_≃_.left-inverse-of A₁≃A₂ x₁))
                                 (_≃_.left-inverse-of A₁≃A₂ x₁))
                          x₂)                                            ≡⟨ cong  p  subst B₂ eq₁ (to (B₁↣B₂ x₁) (subst B₁ p x₂))) $
                                                                                 trans-symˡ _ 
              subst B₂ eq₁ (to (B₁↣B₂ x₁) $ subst B₁ (refl _) x₂)        ≡⟨ cong (subst B₂ eq₁  to (B₁↣B₂ x₁)) $
                                                                                 subst-refl B₁ x₂ 
              subst B₂ eq₁ (to (B₁↣B₂ x₁) x₂)                            ≡⟨ eq₂ ⟩∎
              to (B₁↣B₂ y₁) y₂                                           
        in
        subst B₁ (_≃_.injective A₁≃A₂ eq₁) x₂  ≡⟨ _↣_.injective (B₁↣B₂ y₁) lemma ⟩∎
        y₂                                     ) 
      Σ-≡,≡←≡

∃-preserves-surjections :
   {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
    {B₁ : A₁  Set b₁} {B₂ : A₂  Set b₂}
  (A₁≃A₂ : A₁  A₂)  (∀ x  B₁ x  B₂ (_≃_.to A₁≃A₂ x)) 
  Σ A₁ B₁  Σ A₂ B₂
∃-preserves-surjections {A₁ = A₁} {A₂} {B₁} {B₂} A₁≃A₂ B₁↠B₂ = record
  { logical-equivalence = logical-equivalence′
  ; right-inverse-of    = right-inverse-of′
  }
  where
  open _↠_

  logical-equivalence′ : Σ A₁ B₁  Σ A₂ B₂
  logical-equivalence′ =
    ∃-preserves-logical-equivalences A₁≃A₂ (logical-equivalence  B₁↠B₂)

  abstract
    right-inverse-of′ :
       p 
      _⇔_.to logical-equivalence′ (_⇔_.from logical-equivalence′ p)  p
    right-inverse-of′ = λ p  Σ-≡,≡→≡
      (_≃_.right-inverse-of A₁≃A₂ (proj₁ p))
      (subst B₂ (_≃_.right-inverse-of A₁≃A₂ (proj₁ p))
         (to (B₁↠B₂ _) (from (B₁↠B₂ _)
            (subst B₂ (sym (_≃_.right-inverse-of A₁≃A₂ (proj₁ p)))
               (proj₂ p))))                                         ≡⟨ cong (subst B₂ _) $ right-inverse-of (B₁↠B₂ _) _ 
       subst B₂ (_≃_.right-inverse-of A₁≃A₂ (proj₁ p))
         (subst B₂ (sym (_≃_.right-inverse-of A₁≃A₂ (proj₁ p)))
            (proj₂ p))                                              ≡⟨ subst-subst-sym B₂ _ _ ⟩∎
       proj₂ p )

∃-preserves-bijections :
   {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
    {B₁ : A₁  Set b₁} {B₂ : A₂  Set b₂}
  (A₁≃A₂ : A₁  A₂)  (∀ x  B₁ x  B₂ (_≃_.to A₁≃A₂ x)) 
  Σ A₁ B₁  Σ A₂ B₂
∃-preserves-bijections {A₁ = A₁} {A₂} {B₁} {B₂} A₁≃A₂ B₁↔B₂ = record
  { surjection      = surjection′
  ; left-inverse-of = left-inverse-of′
  }
  where
  open _↔_

  surjection′ : Σ A₁ B₁  Σ A₂ B₂
  surjection′ = ∃-preserves-surjections A₁≃A₂ (surjection  B₁↔B₂)

  abstract
    left-inverse-of′ :
       p  _↠_.from surjection′ (_↠_.to surjection′ p)  p
    left-inverse-of′ = λ p  Σ-≡,≡→≡
      (_≃_.left-inverse-of A₁≃A₂ (proj₁ p))
      (subst B₁ (_≃_.left-inverse-of A₁≃A₂ (proj₁ p))
         (from (B₁↔B₂ _)
            (subst B₂ (sym (_≃_.right-inverse-of A₁≃A₂
                              (_≃_.to A₁≃A₂ (proj₁ p))))
               (to (B₁↔B₂ _) (proj₂ p))))                        ≡⟨ push-subst B₂  x  from (B₁↔B₂ x))
                                                                      (_≃_.left-inverse-of A₁≃A₂ (proj₁ p)) 
       from (B₁↔B₂ _)
         (subst B₂ (cong (_≃_.to A₁≃A₂)
                         (_≃_.left-inverse-of A₁≃A₂ (proj₁ p)))
            (subst B₂ (sym (_≃_.right-inverse-of A₁≃A₂
                              (_≃_.to A₁≃A₂ (proj₁ p))))
               (to (B₁↔B₂ _) (proj₂ p))))                        ≡⟨ cong  eq  from (B₁↔B₂ _)
                                                                                   (subst B₂ eq
                                                                                      (subst B₂ (sym (_≃_.right-inverse-of A₁≃A₂ _))
                                                                                         (to (B₁↔B₂ _) (proj₂ p))))) $
                                                                         _≃_.left-right-lemma A₁≃A₂ _ 
       from (B₁↔B₂ _)
         (subst B₂ (_≃_.right-inverse-of A₁≃A₂
                      (_≃_.to A₁≃A₂ (proj₁ p)))
            (subst B₂ (sym (_≃_.right-inverse-of A₁≃A₂
                              (_≃_.to A₁≃A₂ (proj₁ p))))
               (to (B₁↔B₂ _) (proj₂ p))))                        ≡⟨ cong (from (B₁↔B₂ _)) $ subst-subst-sym B₂ _ _ 
       from (B₁↔B₂ _) (to (B₁↔B₂ _) (proj₂ p))                   ≡⟨ left-inverse-of (B₁↔B₂ _) _ ⟩∎
       proj₂ p                                                   )

-- Σ preserves equivalence.

Σ-preserves :  {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
                {B₁ : A₁  Set b₁} {B₂ : A₂  Set b₂}
              (A₁≃A₂ : A₁  A₂)  (∀ x  B₁ x  B₂ (_≃_.to A₁≃A₂ x)) 
              Σ A₁ B₁  Σ A₂ B₂
Σ-preserves A₁≃A₂ B₁≃B₂ = ↔⇒≃ $
  ∃-preserves-bijections A₁≃A₂ (_≃_.bijection  B₁≃B₂)

-- Π preserves equivalence (assuming extensionality).

Π-preserves :
   {a₁ a₂ b₁ b₂}  Extensionality (a₁  a₂) (b₁  b₂) 
  {A₁ : Set a₁} {A₂ : Set a₂} {B₁ : A₁  Set b₁} {B₂ : A₂  Set b₂} 
  (A₁≃A₂ : A₁  A₂)  (∀ x  B₁ x  B₂ (_≃_.to A₁≃A₂ x)) 
  ((x : A₁)  B₁ x)  ((x : A₂)  B₂ x)
Π-preserves {a₁} {a₂} {b₁} {b₂} ext {A₁} {A₂} {B₁} {B₂} A₁≃A₂ B₁≃B₂ =
  ↔⇒≃ record
    { surjection = record
      { logical-equivalence = record
        { to   = to′
        ; from = from′
        }
      ; right-inverse-of = right-inverse-of′
      }
    ; left-inverse-of = left-inverse-of′
    }
  where
  open _≃_

  to′ : ((x : A₁)  B₁ x)  (x : A₂)  B₂ x
  to′ f x = subst B₂ (right-inverse-of A₁≃A₂ x)
                  (to (B₁≃B₂ (from A₁≃A₂ x))
                      (f (from A₁≃A₂ x)))

  from′ : ((x : A₂)  B₂ x)  (x : A₁)  B₁ x
  from′ f x = from (B₁≃B₂ x) (f (to A₁≃A₂ x))

  abstract
    right-inverse-of′ :  f  to′ (from′ f)  f
    right-inverse-of′ = λ f  lower-extensionality a₁ b₁ ext λ x 
      subst B₂ (right-inverse-of A₁≃A₂ x)
            (to (B₁≃B₂ (from A₁≃A₂ x))
                (from (B₁≃B₂ (from A₁≃A₂ x))
                      (f (to A₁≃A₂ (from A₁≃A₂ x)))))  ≡⟨ cong (subst B₂ (right-inverse-of A₁≃A₂ x)) $
                                                               right-inverse-of (B₁≃B₂ _) _ 
      subst B₂ (right-inverse-of A₁≃A₂ x)
            (f (to A₁≃A₂ (from A₁≃A₂ x)))              ≡⟨ elim  {x y} x≡y  subst B₂ x≡y (f x)  f y)
                                                                x  subst-refl B₂ (f x))
                                                               (right-inverse-of A₁≃A₂ x) ⟩∎
      f x                                              

    left-inverse-of′ :  f  from′ (to′ f)  f
    left-inverse-of′ = λ f  lower-extensionality a₂ b₂ ext λ x 
      from (B₁≃B₂ x)
           (subst B₂ (right-inverse-of A₁≃A₂ (to A₁≃A₂ x))
                  (to (B₁≃B₂ (from A₁≃A₂ (to A₁≃A₂ x)))
                      (f (from A₁≃A₂ (to A₁≃A₂ x)))))             ≡⟨ cong  eq  from (B₁≃B₂ x)
                                                                                    (subst B₂ eq
                                                                                       (to (B₁≃B₂ (from A₁≃A₂ (to A₁≃A₂ x)))
                                                                                          (f (from A₁≃A₂ (to A₁≃A₂ x))))))
                                                                          (sym $ left-right-lemma A₁≃A₂ x) 
      from (B₁≃B₂ x)
           (subst B₂ (cong (to A₁≃A₂) (left-inverse-of A₁≃A₂ x))
                  (to (B₁≃B₂ (from A₁≃A₂ (to A₁≃A₂ x)))
                      (f (from A₁≃A₂ (to A₁≃A₂ x)))))             ≡⟨ sym $ push-subst B₂  x y  from (B₁≃B₂ x) y)
                                                                                      (left-inverse-of A₁≃A₂ x) 
      subst B₁ (left-inverse-of A₁≃A₂ x)
            (from (B₁≃B₂ (from A₁≃A₂ (to A₁≃A₂ x)))
                  (to (B₁≃B₂ (from A₁≃A₂ (to A₁≃A₂ x)))
                      (f (from A₁≃A₂ (to A₁≃A₂ x)))))             ≡⟨ cong (subst B₁ (left-inverse-of A₁≃A₂ x)) $
                                                                       left-inverse-of (B₁≃B₂ _) _ 
      subst B₁ (left-inverse-of A₁≃A₂ x)
            (f (from A₁≃A₂ (to A₁≃A₂ x)))                         ≡⟨ elim  {x y} x≡y  subst B₁ x≡y (f x)  f y)
                                                                           x  subst-refl B₁ (f x))
                                                                          (left-inverse-of A₁≃A₂ x) ⟩∎
      f x                                                         

-- Π preserves equivalence in its second argument (assuming
-- extensionality).
--
-- Note that this proof's "to" component does not use subst, unlike
-- the one in the proof of Π-preserves.

∀-preserves :
   {a b₁ b₂}  Extensionality a (b₁  b₂) 
  {A : Set a} {B₁ : A  Set b₁} {B₂ : A  Set b₂} 
  (∀ x  B₁ x  B₂ x) 
  ((x : A)  B₁ x)  ((x : A)  B₂ x)
∀-preserves {a} {b₁} {b₂} ext {A} {B₁} {B₂} B₁≃B₂ = ↔⇒≃ record
    { surjection = record
      { logical-equivalence = record
        { to   = to′
        ; from = from′
        }
      ; right-inverse-of = right-inverse-of′
      }
    ; left-inverse-of = left-inverse-of′
    }
  where
  open _≃_

  to′ : ((x : A)  B₁ x)  (x : A)  B₂ x
  to′ f x = to (B₁≃B₂ x) (f x)

  from′ : ((x : A)  B₂ x)  (x : A)  B₁ x
  from′ f x = from (B₁≃B₂ x) (f x)

  abstract
    right-inverse-of′ :  f  to′ (from′ f)  f
    right-inverse-of′ = λ f  lower-extensionality a b₁ ext λ x 
      to (B₁≃B₂ x) (from (B₁≃B₂ x) (f x))  ≡⟨ right-inverse-of (B₁≃B₂ x) (f x) ⟩∎
      f x                                  

    left-inverse-of′ :  f  from′ (to′ f)  f
    left-inverse-of′ = λ f  lower-extensionality a b₂ ext λ x 
      from (B₁≃B₂ x) (to (B₁≃B₂ x) (f x))  ≡⟨ left-inverse-of (B₁≃B₂ x) (f x) ⟩∎
      f x                                  

-- Equivalence preserves equivalences (assuming extensionality).

≃-preserves :
   {a₁ a₂ b₁ b₂} 
  Extensionality (a₁  a₂  b₁  b₂) (a₁  a₂  b₁  b₂) 
  {A₁ : Set a₁} {A₂ : Set a₂} {B₁ : Set b₁} {B₂ : Set b₂} 
  A₁  A₂  B₁  B₂  (A₁  B₁)  (A₂  B₂)
≃-preserves {a₁} {a₂} {b₁} {b₂} ext {A₁} {A₂} {B₁} {B₂} A₁≃A₂ B₁≃B₂ =
  ↔⇒≃ (record
    { surjection = record
      { logical-equivalence = record
        { to   = λ A₁≃B₁  B₁≃B₂  A₁≃B₁  inverse A₁≃A₂
        ; from = λ A₂≃B₂  inverse B₁≃B₂  A₂≃B₂  A₁≃A₂
        }
      ; right-inverse-of = to∘from
      }
    ; left-inverse-of = from∘to
    })
  where
  open _≃_

  abstract
    to∘from :
      (A₂≃B₂ : A₂  B₂) 
      B₁≃B₂  (inverse B₁≃B₂  A₂≃B₂  A₁≃A₂)  inverse A₁≃A₂  A₂≃B₂
    to∘from A₂≃B₂ =
      lift-equality (lower-extensionality (a₁  b₁) (a₁  b₁) ext) $
        lower-extensionality (a₁  b₁  b₂) (a₁  a₂  b₁) ext λ x 
          to B₁≃B₂ (from B₁≃B₂ (to A₂≃B₂ (to A₁≃A₂ (from A₁≃A₂ x))))  ≡⟨ right-inverse-of B₁≃B₂ _ 
          to A₂≃B₂ (to A₁≃A₂ (from A₁≃A₂ x))                          ≡⟨ cong (to A₂≃B₂) $ right-inverse-of A₁≃A₂ _ ⟩∎
          to A₂≃B₂ x                                                  

    from∘to :
      (A₁≃B₁ : A₁  B₁) 
      inverse B₁≃B₂  (B₁≃B₂  A₁≃B₁  inverse A₁≃A₂)  A₁≃A₂  A₁≃B₁
    from∘to A₁≃B₁ =
      lift-equality (lower-extensionality (a₂  b₂) (a₂  b₂) ext) $
        lower-extensionality (a₂  b₁  b₂) (a₁  a₂  b₂) ext λ x 
          from B₁≃B₂ (to B₁≃B₂ (to A₁≃B₁ (from A₁≃A₂ (to A₁≃A₂ x))))  ≡⟨ left-inverse-of B₁≃B₂ _ 
          to A₁≃B₁ (from A₁≃A₂ (to A₁≃A₂ x))                          ≡⟨ cong (to A₁≃B₁) $ left-inverse-of A₁≃A₂ _ ⟩∎
          to A₁≃B₁ x                                                  

-- Equivalence preserves bijections (assuming extensionality).

≃-preserves-bijections :
   {a₁ a₂ b₁ b₂} 
  Extensionality (a₁  a₂  b₁  b₂) (a₁  a₂  b₁  b₂) 
  {A₁ : Set a₁} {A₂ : Set a₂} {B₁ : Set b₁} {B₂ : Set b₂} 
  A₁  A₂  B₁  B₂  (A₁  B₁)  (A₂  B₂)
≃-preserves-bijections ext A₁↔A₂ B₁↔B₂ =
  _≃_.bijection $ ≃-preserves ext (↔⇒≃ A₁↔A₂) (↔⇒≃ B₁↔B₂)

------------------------------------------------------------------------
-- Another property

abstract

  -- As a consequence of extensionality-isomorphism and ≃-≡ we get a
  -- strengthening of W-≡,≡↠≡.

  W-≡,≡≃≡ :  {a b} {A : Set a} {B : A  Set b} 
            (∀ {x} {C : B x  Set (a  b)}  Extensionality′ (B x) C) 
             {x y} {f : B x  W A B} {g : B y  W A B} 
            ( λ (p : x  y)   i  f i  g (subst B p i)) 
            (sup x f  sup y g)
  W-≡,≡≃≡ {a} {A = A} {B} ext {x} {y} {f} {g} =
    ( λ p   i  f i  g (subst B p i))        ≃⟨ Σ-preserves id lemma 
    ( λ p  subst  x  B x  W A B) p f  g)  ≃⟨ ↔⇒≃ Σ-≡,≡↔≡ 
    ((x , f)  (y , g))                          ≃⟨ ≃-≡ (↔⇒≃ W-unfolding) ⟩□
    (sup x f  sup y g)                          
    where
    lemma : (p : x  y) 
            (∀ i  f i  g (subst B p i)) 
            (subst  x  B x  W A B) p f  g)
    lemma p = elim
       {x y} p  (f : B x  W A B) (g : B y  W A B) 
                   (∀ i  f i  g (subst B p i)) 
                   (subst  x  B x  W A B) p f  g))
       x f g 
         (∀ i  f i  g (subst B (refl x) i))        ≃⟨ subst  h  (∀ i  f i  g (h i))  (∀ i  f i  g i))
                                                              (sym (lower-extensionality₂ a ext (subst-refl B)))
                                                              id 
         (∀ i  f i  g i)                           ≃⟨ extensionality-isomorphism ext 
         (f  g)                                     ≃⟨ subst  h  (f  g)  (h  g))
                                                              (sym $ subst-refl  x'  B x'  W A B) f)
                                                              id ⟩□
         (subst  x  B x  W A B) (refl x) f  g)  )
      p f g