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

{-# OPTIONS --cubical-compatible --safe #-}

open import Equality

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

open import Bijection eq as Bijection using (_↔_; Has-quasi-inverse)
open Derived-definitions-and-properties eq
open import Embedding eq as Emb using (Is-embedding; Embedding)
open import Equality.Decidable-UIP eq
open import Equality.Decision-procedures eq
open import Equivalence eq as Eq using (_≃_; module _≃_; Is-equivalence)
import Equivalence.Contractible-preimages eq as CP
open import Equivalence.Erased.Basics eq as EEq using (_≃ᴱ_)
import Equivalence.Half-adjoint eq as HA
open import Erased.Basics as E using (Erased)
open import Extensionality eq
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 as L using (_⇔_; module _⇔_)
open import Nat eq hiding (_≟_)
open import Preimage eq as Preimage using (_⁻¹_)
open import Prelude as P hiding (id) renaming (_∘_ to _⊚_)
open import Surjection eq as Surjection using (_↠_; Split-surjective)

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

-- The universe includes implications, logical equivalences,
-- injections, embeddings, surjections, bijections, equivalences, and
-- equivalences with erased proofs.

data Kind : Type where
  implication
    logical-equivalence
    injection
    embedding
    surjection
    bijection
    equivalence
    equivalenceᴱ : Kind

-- The interpretation of the universe.

infix 0 _↝[_]_

_↝[_]_ :  {ℓ₁ ℓ₂}  Type ℓ₁  Kind  Type ℓ₂  Type _
A ↝[ implication         ] B = A  B
A ↝[ logical-equivalence ] B = A  B
A ↝[ injection           ] B = A  B
A ↝[ embedding           ] B = Embedding A B
A ↝[ surjection          ] B = A  B
A ↝[ bijection           ] B = A  B
A ↝[ equivalence         ] B = A  B
A ↝[ equivalenceᴱ        ] B = A ≃ᴱ B

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

from-equivalence :  {k a b} {A : Type a} {B : Type b} 
                   A  B  A ↝[ k ] B
from-equivalence {implication}         = _≃_.to
from-equivalence {logical-equivalence} = _≃_.logical-equivalence
from-equivalence {injection}           = _≃_.injection
from-equivalence {embedding}           = Emb.≃→Embedding
from-equivalence {surjection}          = _≃_.surjection
from-equivalence {bijection}           = _≃_.bijection
from-equivalence {equivalence}         = P.id
from-equivalence {equivalenceᴱ}        = EEq.≃→≃ᴱ

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

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

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

to-implication :  {k a b} {@0 A : Type a} {@0 B : Type b} 
                 A ↝[ k ] B  A  B
to-implication {implication} f =
  f
to-implication {logical-equivalence} f =
  let record { to = to } = f in to
to-implication {injection} f =
  let record { to = to } = f in to
to-implication {embedding} f =
  let record { to = to } = f in to
to-implication {surjection} f =
  let record
        { logical-equivalence = record
          { to = to
          }
        } = f
  in to
to-implication {bijection} f =
  let record
        { surjection = record
          { logical-equivalence = record
            { to = to
            }
          }
        } = f
  in to
to-implication {equivalence} f =
  let record { to = to } = f in to
to-implication {equivalenceᴱ} f =
  _≃ᴱ_.to f

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

data Symmetric-kind : Type where
  logical-equivalence bijection equivalence equivalenceᴱ :
    Symmetric-kind

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

inverse :  {k a b} {A : Type a} {B : Type b} 
          A ↝[  k ⌋-sym ] B  B ↝[  k ⌋-sym ] A
inverse {logical-equivalence} = L.inverse
inverse {bijection}           = Bijection.inverse
inverse {equivalence}         = Eq.inverse
inverse {equivalenceᴱ}        = EEq.inverse

-- If there is a symmetric kind of function from A to B, then A and B
-- are logically equivalent.

sym→⇔ :
   {k a b} {A : Type a} {B : Type b} 
  A ↝[  k ⌋-sym ] B  A  B
sym→⇔ {k = logical-equivalence} = P.id
sym→⇔ {k = bijection}           = from-bijection
sym→⇔ {k = equivalence}         = from-equivalence
sym→⇔ {k = equivalenceᴱ}        = _≃ᴱ_.logical-equivalence

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

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

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

infix 0 _↔[_]_

_↔[_]_ :  {ℓ₁ ℓ₂}  Type ℓ₁  Isomorphism-kind  Type ℓ₂  Type _
A ↔[ k ] B = A ↝[  k ⌋-iso ] B

from-isomorphism :  {k₁ k₂ a b} {A : Type a} {B : Type 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 : Type a} {B : Type 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   embedding           = refl _
  t∘f bijection   surjection          = refl _
  t∘f bijection   bijection           = refl _
  t∘f bijection   equivalence         = refl _
  t∘f bijection   equivalenceᴱ        = refl _
  t∘f equivalence implication         = refl _
  t∘f equivalence logical-equivalence = refl _
  t∘f equivalence injection           = refl _
  t∘f equivalence embedding           = refl _
  t∘f equivalence surjection          = refl _
  t∘f equivalence bijection           = refl _
  t∘f equivalence equivalence         = refl _
  t∘f equivalence equivalenceᴱ        = refl _

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

-- All the different kinds of functions form preorders.

-- Composition.

infixr 9 _∘_

_∘_ :  {k a b c} {A : Type a} {B : Type b} {C : Type c} 
      B ↝[ k ] C  A ↝[ k ] B  A ↝[ k ] C
_∘_ {implication}         = λ f g  f  g
_∘_ {logical-equivalence} = L._∘_
_∘_ {injection}           = Injection._∘_
_∘_ {embedding}           = Emb._∘_
_∘_ {surjection}          = Surjection._∘_
_∘_ {bijection}           = Bijection._∘_
_∘_ {equivalence}         = Eq._∘_
_∘_ {equivalenceᴱ}        = EEq._∘_

-- Identity.

id :  {k a} {A : Type a}  A ↝[ k ] A
id {implication}         = P.id
id {logical-equivalence} = L.id
id {injection}           = Injection.id
id {embedding}           = Emb.id
id {surjection}          = Surjection.id
id {bijection}           = Bijection.id
id {equivalence}         = Eq.id
id {equivalenceᴱ}        = EEq.id

-- "Equational" reasoning combinators.

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

-- For an explanation of why step-↝ and step-↔ are defined in this
-- way, see Equality.step-≡.

step-↝ :  {k a b c} (A : Type a) {B : Type b} {C : Type c} 
         B ↝[ k ] C  A ↝[ k ] B  A ↝[ k ] C
step-↝ _ = _∘_

syntax step-↝ A B↝C A↝B = A ↝⟨ A↝B  B↝C

step-↔ :  {k₁ k₂ a b c} (A : Type a) {B : Type b} {C : Type c} 
         B ↝[ k₂ ] C  A ↔[ k₁ ] B  A ↝[ k₂ ] C
step-↔ _ B↝C A↔B = step-↝ _ B↝C (from-isomorphism A↔B)

syntax step-↔ A B↝C A↔B = A ↔⟨ A↔B  B↝C

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

_□ :  {k a} (A : Type a)  A ↝[ k ] A
A  = id

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

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

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

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

$⟨_⟩_ :  {k a b} {@0 A : Type a} {@0 B : Type 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 : Type 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 embedding           = refl _
to-implication-id surjection          = refl _
to-implication-id bijection           = refl _
to-implication-id equivalence         = refl _
to-implication-id equivalenceᴱ        = refl _

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

to-implication-∘ :
   {a b c} {A : Type a} {B : Type b} {C : Type 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-∘ embedding           = refl _
to-implication-∘ surjection          = refl _
to-implication-∘ bijection           = refl _
to-implication-∘ equivalence         = refl _
to-implication-∘ equivalenceᴱ        = refl _

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

to-implication-inverse-id :
   {a} {A : Type 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 _
to-implication-inverse-id equivalenceᴱ        = refl _

------------------------------------------------------------------------
-- Conditional extensionality

-- Code that provides support for proving general statements about
-- functions of different kinds, in which the statements involve
-- assumptions of extensionality for some kinds of functions, but not
-- all. For some examples, see ∀-cong and ∀-intro.

-- Kinds for which extensionality is not provided.

data Without-extensionality : Type where
  implication logical-equivalence : Without-extensionality

⌊_⌋-without : Without-extensionality  Kind
 implication         ⌋-without = implication
 logical-equivalence ⌋-without = logical-equivalence

-- Kinds for which erased extensionality is not provided.

data With-erased-extensionality : Type where
  equivalenceᴱ : With-erased-extensionality

⌊_⌋-with-erased : With-erased-extensionality  Kind
 equivalenceᴱ ⌋-with-erased = equivalenceᴱ

-- Kinds for which extensionality is provided.

data With-extensionality : Type where
  injection embedding surjection bijection equivalence :
    With-extensionality

⌊_⌋-with : With-extensionality  Kind
 injection   ⌋-with = injection
 embedding   ⌋-with = embedding
 surjection  ⌋-with = surjection
 bijection   ⌋-with = bijection
 equivalence ⌋-with = equivalence

-- Kinds annotated with information about what kind of extensionality
-- is provided, if any.

data Extensionality-kind : Kind  Type where
  without-extensionality     : (k : Without-extensionality) 
                               Extensionality-kind  k ⌋-without
  with-erased-extensionality : (k : With-erased-extensionality) 
                               Extensionality-kind  k ⌋-with-erased
  with-extensionality        : (k : With-extensionality) 
                               Extensionality-kind  k ⌋-with

-- Is extensionality provided for the given kind?

extensionality? : (k : Kind)  Extensionality-kind k
extensionality? implication         = without-extensionality implication
extensionality? logical-equivalence = without-extensionality
                                        logical-equivalence
extensionality? injection           = with-extensionality injection
extensionality? embedding           = with-extensionality embedding
extensionality? surjection          = with-extensionality surjection
extensionality? bijection           = with-extensionality bijection
extensionality? equivalence         = with-extensionality equivalence
extensionality? equivalenceᴱ        = with-erased-extensionality
                                        equivalenceᴱ

-- Extensionality, but only for certain kinds of functions, and
-- possibly erased.

Extensionality? : Kind  (a b : Level)  Type (lsuc (a  b))
Extensionality? k a b with extensionality? k
... | without-extensionality _     =  _ 
... | with-erased-extensionality _ = Erased (Extensionality a b)
... | with-extensionality _        = Extensionality a b

-- A variant of _↝[_]_. A ↝[ c ∣ d ] B means that A ↝[ k ] B can be
-- proved for all kinds k, in some cases assuming extensionality (for
-- the levels c and d).

infix 0 _↝[_∣_]_

_↝[_∣_]_ :
   {a b} 
  Type a  (c d : Level)  Type b  Type (a  b  lsuc (c  d))
A ↝[ c  d ] B =  {k}  Extensionality? k c d  A ↝[ k ] B

-- A variant of _↝[_∣_]_ with erased extensionality assumptions.

infix 0 _↝[_∣_]ᴱ_

_↝[_∣_]ᴱ_ :
   {a b} 
  Type a  (c d : Level)  Type b  Type (a  b  lsuc (c  d))
A ↝[ c  d ]ᴱ B =  {k}  @0 Extensionality? k c d  A ↝[ k ] B

-- Turns extensionality into conditional extensionality.

forget-ext? :  k {a b}  Extensionality a b  Extensionality? k a b
forget-ext? k with extensionality? k
... | without-extensionality _     = _
... | with-erased-extensionality _ = E.[_]→
... | with-extensionality _        = id

-- A variant of lower-extensionality.

lower-extensionality? :
   k {a b}   
  Extensionality? k (a  ) (b  )  Extensionality? k a b
lower-extensionality? k   with extensionality? k
... | without-extensionality _     = _
... | with-erased-extensionality _ = E.map (lower-extensionality  )
... | with-extensionality _        = lower-extensionality  

-- Some functions that can be used to generalise results.

generalise-ext? :
   {a b c d} {A : Type a} {B : Type b} 
  (A⇔B : A  B) 
  (Extensionality c d 
   let open _⇔_ A⇔B in
   (∀ x  to (from x)  x) ×
   (∀ x  from (to x)  x)) 
  A ↝[ c  d ] B
generalise-ext? A⇔B hyp {k = k} with extensionality? k
... | without-extensionality implication =
  λ _  _⇔_.to A⇔B
... | without-extensionality logical-equivalence =
  λ _  A⇔B
... | with-extensionality _ = λ ext 
  from-bijection record
    { surjection = record
      { logical-equivalence = A⇔B
      ; right-inverse-of    = hyp ext .proj₁
      }
    ; left-inverse-of = hyp ext .proj₂
    }
... | with-erased-extensionality equivalenceᴱ = λ (E.[ ext ]) 
  EEq.↔→≃ᴱ (_⇔_.to A⇔B) (_⇔_.from A⇔B) (hyp ext .proj₁) (hyp ext .proj₂)

generalise-ext?′ :
   {a b c d} {A : Type a} {B : Type b} 
  A  B 
  (Extensionality c d  A  B) 
  (@0 Extensionality c d  A ≃ᴱ B) 
  A ↝[ c  d ] B
generalise-ext?′ f⇔ f↔ f≃ᴱ {k = k} with extensionality? k
... | without-extensionality implication =
  λ _  _⇔_.to f⇔
... | without-extensionality logical-equivalence =
  λ _  f⇔
... | with-extensionality _ = λ ext 
  from-isomorphism (f↔ ext)
... | with-erased-extensionality equivalenceᴱ = λ ext 
  f≃ᴱ (E.erased ext)

generalise-erased-ext? :
   {a b c d} {A : Type a} {B : Type b} 
  A  B 
  (@0 Extensionality c d  A  B) 
  A ↝[ c  d ]ᴱ B
generalise-erased-ext? f⇔ f↔ {k = k} with extensionality? k
... | without-extensionality implication =
  λ _  _⇔_.to f⇔
... | without-extensionality logical-equivalence =
  λ _  f⇔
... | with-extensionality _ = λ ext 
  from-isomorphism (f↔ ext)
... | with-erased-extensionality _ = λ ext 
  from-isomorphism (f↔ (E.erased ext))

generalise-ext?-prop :
   {a b c d} {A : Type a} {B : Type b} 
  A  B 
  (Extensionality c d  Is-proposition A) 
  (Extensionality c d  Is-proposition B) 
  A ↝[ c  d ] B
generalise-ext?-prop f⇔ A-prop B-prop =
  generalise-ext?
    f⇔
     ext 
          _  B-prop ext _ _)
       ,  _  A-prop ext _ _))

generalise-erased-ext?-prop :
   {a b c d} {A : Type a} {B : Type b} 
  A  B 
  (@0 Extensionality c d  Is-proposition A) 
  (@0 Extensionality c d  Is-proposition B) 
  A ↝[ c  d ]ᴱ B
generalise-erased-ext?-prop f⇔ A-prop B-prop =
  generalise-erased-ext?
    f⇔
     ext  _≃_.bijection $
               _↠_.from (Eq.≃↠⇔ (A-prop ext) (B-prop ext)) f⇔)

generalise-ext?-sym :
   {a b c d} {A : Type a} {B : Type b} 
  (∀ {k}  Extensionality?  k ⌋-sym c d  A ↝[  k ⌋-sym ] B) 
  A ↝[ c  d ] B
generalise-ext?-sym hyp {k = k} ext with extensionality? k
... | without-extensionality implication =
  _⇔_.to $ hyp {k = logical-equivalence} ext
... | without-extensionality logical-equivalence =
  hyp {k = logical-equivalence} ext
... | with-extensionality _ =
  from-bijection $ hyp {k = bijection} ext
... | with-erased-extensionality equivalenceᴱ =
  hyp {k = equivalenceᴱ} ext

generalise-erased-ext?-sym :
   {a b c d} {A : Type a} {B : Type b} 
  (∀ {k}  @0 Extensionality?  k ⌋-sym c d  A ↝[  k ⌋-sym ] B) 
  A ↝[ c  d ]ᴱ B
generalise-erased-ext?-sym hyp = generalise-erased-ext? (hyp _) hyp

-- General results of the kind produced by generalise-ext? are
-- symmetric.

inverse-ext? :
   {a b c d} {A : Type a} {B : Type b} 
  A ↝[ c  d ] B  B ↝[ c  d ] A
inverse-ext? hyp = generalise-ext?-sym (inverse  hyp)

inverse-erased-ext? :
   {a b c d} {A : Type a} {B : Type b} 
  A ↝[ c  d ]ᴱ B  B ↝[ c  d ]ᴱ A
inverse-erased-ext? hyp =
  generalise-erased-ext?-sym  ext  inverse (hyp ext))

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

  ≡⇒↝-sym :  k {} {A B : Type } {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 : Type } {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 : Type   Type 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 ≡⇒↝ in terms of subst.

  ≡⇒↝-in-terms-of-subst :
     k {} {A B : Type } (A≡B : A  B) 
    ≡⇒↝ k A≡B  subst (A ↝[ k ]_) A≡B id
  ≡⇒↝-in-terms-of-subst k {B = B} = elim₁
     {A} A≡B  ≡⇒↝ k A≡B  subst (A ↝[ k ]_) A≡B id)
    (≡⇒↝ k (refl B)                 ≡⟨ ≡⇒↝-refl 
     id                             ≡⟨ sym $ subst-refl _ _ ⟩∎
     subst (B ↝[ k ]_) (refl B) id  )

  ≡⇒↝-in-terms-of-subst-sym :
     k {} {A B : Type } (A≡B : A  B) 
    ≡⇒↝ k A≡B  subst (_↝[ k ] B) (sym A≡B) id
  ≡⇒↝-in-terms-of-subst-sym k {B = B} = elim₁
     {A} A≡B  ≡⇒↝ k A≡B  subst (_↝[ k ] B) (sym A≡B) id)
    (≡⇒↝ k (refl B)                       ≡⟨ ≡⇒↝-refl 
     id                                   ≡⟨ sym $ subst-refl _ _ 
     subst (_↝[ k ] B) (refl B) id        ≡⟨ cong (flip (subst _) _) $ sym sym-refl ⟩∎
     subst (_↝[ k ] B) (sym (refl B)) id  )

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

  subst-in-terms-of-≡⇒↝ :
     k {a p} {A : Type a} {x y} (x≡y : x  y) (P : A  Type 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 : Type a} {x y} (x≡y : x  y) (P : A  Type 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  

  -- A variant of subst-in-terms-of-≡⇒↝ for cong₂.

  ≡⇒↝-cong₂≡subst-subst :
     k {a b p} {A : Type a} {B : Type b} {x y u v}
      {x≡y : x  y} {u≡v : u  v} {P : A  B  Type p} {p} 
    to-implication (≡⇒↝ k (cong₂ P x≡y u≡v)) p 
    subst (P _) u≡v (subst (flip P _) x≡y p)
  ≡⇒↝-cong₂≡subst-subst k {x≡y = x≡y} {u≡v = u≡v} {P = P} {p = p} =
    to-implication (≡⇒↝ k (cong₂ P x≡y u≡v)) p                        ≡⟨⟩

    to-implication
      (≡⇒↝ k (trans (cong (flip P _) x≡y) (cong (P _) u≡v))) p        ≡⟨ cong (_$ p) $ ≡⇒↝-trans k 

    to-implication
      (≡⇒↝ k (cong (P _) u≡v)  ≡⇒↝ k (cong (flip P _) x≡y)) p        ≡⟨ cong (_$ p) $ to-implication-∘ k 

    to-implication (≡⇒↝ k (cong (P _) u≡v))
      (to-implication (≡⇒↝ k (cong (flip P _) x≡y)) p)                ≡⟨ sym $ subst-in-terms-of-≡⇒↝ k _ _ _ 

    subst (P _) u≡v (to-implication (≡⇒↝ k (cong (flip P _) x≡y)) p)  ≡⟨ cong (subst (P _) u≡v) $ sym $
                                                                         subst-in-terms-of-≡⇒↝ k _ _ _ ⟩∎
    subst (P _) u≡v (subst (flip P _) x≡y p)                          

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

  subst-id-in-terms-of-≡⇒↝ :
     k {a} {A B : Type a} {A≡B : A  B} {x} 
    subst id A≡B x  to-implication (≡⇒↝ k A≡B) x
  subst-id-in-terms-of-≡⇒↝ k {A≡B = A≡B} {x = x} =
    subst id A≡B x                          ≡⟨ subst-in-terms-of-≡⇒↝ k _ _ _ 
    to-implication (≡⇒↝ k (cong id A≡B)) x  ≡⟨ cong  eq  to-implication (≡⇒↝ k eq) x) $ sym $ cong-id _ ⟩∎
    to-implication (≡⇒↝ k A≡B) x            

  subst-id-in-terms-of-inverse∘≡⇒↝ :
     k {a} {A B : Type a} {A≡B : A  B} {y} 
    subst id (sym A≡B) y 
    to-implication (inverse (≡⇒↝  k ⌋-sym A≡B)) y
  subst-id-in-terms-of-inverse∘≡⇒↝ k {A≡B = A≡B} {y = y} =
    subst id (sym A≡B) y                                      ≡⟨ subst-in-terms-of-inverse∘≡⇒↝ k _ _ _ 
    to-implication (inverse (≡⇒↝  k ⌋-sym (cong id A≡B))) y  ≡⟨ cong  eq  to-implication (inverse (≡⇒↝  k ⌋-sym eq)) y) $ sym $ cong-id _ ⟩∎
    to-implication (inverse (≡⇒↝  k ⌋-sym A≡B)) y            

  to-implication-≡⇒↝ :
     k {} {A B : Type } (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)         )

------------------------------------------------------------------------
-- One can replace the "to" and "from" functions with extensionally
-- equal functions

-- One can replace the "to" function with an extensionally equal
-- function.

with-other-function :
   {k a b} {A : Type a} {B : Type b}
  (A↝B : A ↝[ k ] B) (f : A  B) 
  (∀ x  to-implication A↝B x  f x) 
  A ↝[ k ] B
with-other-function {k = implication} _ f _ = f

with-other-function {k = logical-equivalence} A⇔B f _ = record
  { to   = f
  ; from = _⇔_.from A⇔B
  }

with-other-function {k = injection} A↣B f ≡f = record
  { to        = f
  ; injective = λ {x = x} {y = y} 
      f x  f y                    →⟨ flip trans (sym $ ≡f y)  trans (≡f x) 
      _↣_.to A↣B x  _↣_.to A↣B y  →⟨ _↣_.injective A↣B ⟩□
      x  y                        
  }

with-other-function {k = embedding} A↣B f ≡f = record
  { to           = f
  ; is-embedding = λ x y 
      _≃_.is-equivalence $
      Eq.with-other-function
        (x  y                                    ↝⟨ Eq.⟨ _ , Embedding.is-embedding A↣B x y  
         Embedding.to A↣B x  Embedding.to A↣B y  ↝⟨ ≡⇒↝ _ $ cong₂ _≡_ (≡f x) (≡f y) ⟩□
         f x  f y                                )
        (cong f)
        (elim
            {x = x} {y = y} x≡y 
              _≃_.to (≡⇒↝ _ (cong₂ _≡_ (≡f x) (≡f y)))
                (cong (Embedding.to A↣B) x≡y) 
              cong f x≡y)
            x 
              _≃_.to (≡⇒↝ _ (cong₂ _≡_ (≡f x) (≡f x)))
                (cong (Embedding.to A↣B) (refl _))                ≡⟨ cong (_≃_.to (≡⇒↝ _ _)) $
                                                                     cong-refl _ 

              _≃_.to (≡⇒↝ _ (cong₂ _≡_ (≡f x) (≡f x))) (refl _)   ≡⟨ cong (_$ _) $
                                                                     ≡⇒↝-trans equivalence 
              _≃_.to (≡⇒↝ _ (cong (_ ≡_) (≡f x)))
                (_≃_.to (≡⇒↝ _ (cong (_≡ _) (≡f x))) (refl _))    ≡⟨ sym $
                                                                     trans (subst-in-terms-of-≡⇒↝ equivalence _ _ _) $
                                                                     cong (_≃_.to (≡⇒↝ _ _)) $
                                                                     subst-in-terms-of-≡⇒↝ equivalence _ _ _ 

              subst (_ ≡_) (≡f x) (subst (_≡ _) (≡f x) (refl _))  ≡⟨ trans (cong (subst (_ ≡_) (≡f x)) $
                                                                            trans subst-trans-sym $
                                                                            trans-reflʳ _) $
                                                                     sym trans-subst 

              trans (sym (≡f x)) (≡f x)                           ≡⟨ trans-symˡ _ 

              refl (f x)                                          ≡⟨ sym $ cong-refl _ ⟩∎

              cong f (refl x)                                     ))
  }

with-other-function {k = surjection} A↠B f ≡f = record
  { logical-equivalence = record
    { to   = f
    ; from = _↠_.from A↠B
    }
  ; right-inverse-of = λ x 
      f (_↠_.from A↠B x)           ≡⟨ sym $ ≡f _ 
      _↠_.to A↠B (_↠_.from A↠B x)  ≡⟨ _↠_.right-inverse-of A↠B _ ⟩∎
      x                            
  }

with-other-function {k = bijection} = Bijection.with-other-function

with-other-function {k = equivalence} = Eq.with-other-function

with-other-function {k = equivalenceᴱ} A≃ᴱB f ≡f =
  EEq.with-other-function A≃ᴱB f ≡f

-- The function with-other-function changes the "to" function in the
-- advertised way.

to-implication-with-other-function :
   k {a b} {A : Type a} {B : Type b} {A↝B : A ↝[ k ] B} {f : A  B}
    {≡f :  x  to-implication A↝B x  f x} {x} 
  to-implication (with-other-function A↝B f ≡f) x  f x
to-implication-with-other-function = λ where
  implication          refl _
  logical-equivalence  refl _
  injection            refl _
  embedding            refl _
  surjection           refl _
  bijection            refl _
  equivalence          refl _
  equivalenceᴱ         refl _

-- The function with-other-function does not change the "from"
-- function (if any).

to-implication-inverse-with-other-function :
   k {a b} {A : Type a} {B : Type b} {A↝B : A ↝[  k ⌋-sym ] B}
    {f : A  B} {≡f :  x  to-implication A↝B x  f x} {x} 
  to-implication (inverse (with-other-function A↝B f ≡f)) x 
  to-implication (inverse A↝B) x
to-implication-inverse-with-other-function = λ where
  logical-equivalence  refl _
  bijection            refl _
  equivalence          refl _
  equivalenceᴱ         refl _

-- A variant of with-other-function.

with-other-function-ext? :
   {k a b c d} {A : Type a} {B : Type b}
  (A↝B : A ↝[ k ] B) (f : A  B) 
  (Extensionality c d   x  to-implication A↝B x  f x) 
  Extensionality? k c d  A ↝[ k ] B
with-other-function-ext? {k = implication} _ f _ _ =
  f
with-other-function-ext? {k = logical-equivalence} A⇔B f _ _ =
  record A⇔B { to = f }
with-other-function-ext? {k = injection} A↣B f ≡f ext =
  with-other-function A↣B f (≡f ext)
with-other-function-ext? {k = embedding} A↣B f ≡f ext =
  with-other-function A↣B f (≡f ext)
with-other-function-ext? {k = surjection} A↠B f ≡f ext =
  with-other-function A↠B f (≡f ext)
with-other-function-ext? {k = bijection} A↔B f ≡f ext =
  with-other-function A↔B f (≡f ext)
with-other-function-ext? {k = equivalence} A≃B f ≡f ext =
  with-other-function A≃B f (≡f ext)
with-other-function-ext? {k = equivalenceᴱ} A≃ᴱB f ≡f ext =
  EEq.with-other-function A≃ᴱB f (≡f (ext .E.erased))

-- The function with-other-function-ext? changes the "to" function in
-- the correct way.

to-implication-with-other-function-ext? :
   k {a b c d} {A : Type a} {B : Type b} {A↝B : A ↝[ k ] B} {f : A  B}
    {≡f : Extensionality c d   x  to-implication A↝B x  f x}
    {x ext} 
  to-implication (with-other-function-ext? A↝B f ≡f ext) x  f x
to-implication-with-other-function-ext? = λ where
  implication          refl _
  logical-equivalence  refl _
  injection            refl _
  embedding            refl _
  surjection           refl _
  bijection            refl _
  equivalence          refl _
  equivalenceᴱ         refl _

-- The function with-other-function-ext? does not change the "from"
-- function (if any).

to-implication-inverse-with-other-function-ext? :
   k {a b c d} {A : Type a} {B : Type b} {A↝B : A ↝[  k ⌋-sym ] B}
    {f : A  B}
    {≡f : Extensionality c d   x  to-implication A↝B x  f x}
    {x ext} 
  to-implication (inverse (with-other-function-ext? A↝B f ≡f ext)) x 
  to-implication (inverse A↝B) x
to-implication-inverse-with-other-function-ext? = λ where
  logical-equivalence  refl _
  bijection            refl _
  equivalence          refl _
  equivalenceᴱ         refl _

-- One can replace the "from" function with an extensionally equal
-- function.

with-other-inverse :
   {k a b} {A : Type a} {B : Type b}
  (A↝B : A ↝[  k ⌋-sym ] B) (f : B  A) 
  (∀ x  to-implication (inverse A↝B) x  f x) 
  A ↝[  k ⌋-sym ] B
with-other-inverse {k = logical-equivalence} A⇔B f _ = record
  { to   = _⇔_.to A⇔B
  ; from = f
  }

with-other-inverse {k = bijection} = Bijection.with-other-inverse

with-other-inverse {k = equivalence} = Eq.with-other-inverse

with-other-inverse {k = equivalenceᴱ} A≃ᴱB f ≡f =
  EEq.with-other-inverse A≃ᴱB f ≡f

-- The function with-other-inverse does not change the "to" function.

to-implication-with-other-inverse :
   k {a b} {A : Type a} {B : Type b} {A↝B : A ↝[  k ⌋-sym ] B}
    {f : B  A} {≡f :  x  to-implication (inverse A↝B) x  f x} {x} 
  to-implication (with-other-inverse A↝B f ≡f) x 
  to-implication A↝B x
to-implication-with-other-inverse = λ where
  logical-equivalence  refl _
  bijection            refl _
  equivalence          refl _
  equivalenceᴱ         refl _

-- The function with-other-inverse changes the "from" function in the
-- advertised way.

to-implication-inverse-with-other-inverse :
   k {a b} {A : Type a} {B : Type b} {A↝B : A ↝[  k ⌋-sym ] B}
    {f : B  A} {≡f :  x  to-implication (inverse A↝B) x  f x} {x} 
  to-implication (inverse (with-other-inverse A↝B f ≡f)) x  f x
to-implication-inverse-with-other-inverse = λ where
  logical-equivalence  refl _
  bijection            refl _
  equivalence          refl _
  equivalenceᴱ         refl _

-- A variant of with-other-inverse.

with-other-inverse-ext? :
   {k a b c d} {A : Type a} {B : Type b}
  (A↝B : A ↝[  k ⌋-sym ] B) (f : B  A) 
  (Extensionality c d   x  to-implication (inverse A↝B) x  f x) 
  Extensionality?  k ⌋-sym c d  A ↝[  k ⌋-sym ] B
with-other-inverse-ext? {k = logical-equivalence} A⇔B f _ _ =
  record A⇔B { from = f }
with-other-inverse-ext? {k = bijection} A↔B f ≡f ext =
  with-other-inverse A↔B f (≡f ext)
with-other-inverse-ext? {k = equivalence} A≃B f ≡f ext =
  with-other-inverse A≃B f (≡f ext)
with-other-inverse-ext? {k = equivalenceᴱ} A≃ᴱB f ≡f ext =
  EEq.with-other-inverse A≃ᴱB f (≡f (ext .E.erased))

-- The function with-other-inverse-ext? does not change the "to"
-- function.

to-implication-with-other-inverse-ext? :
   k {a b c d} {A : Type a} {B : Type b} {A↝B : A ↝[  k ⌋-sym ] B}
    {f : B  A}
    {≡f : Extensionality c d 
           x  to-implication (inverse A↝B) x  f x}
    {x ext} 
  to-implication (with-other-inverse-ext? A↝B f ≡f ext) x 
  to-implication A↝B x
to-implication-with-other-inverse-ext? = λ where
  logical-equivalence  refl _
  bijection            refl _
  equivalence          refl _
  equivalenceᴱ         refl _

-- The function with-other-inverse-ext? changes the "from" function in
-- the correct way.

to-implication-inverse-with-other-inverse-ext? :
   k {a b c d} {A : Type a} {B : Type b} {A↝B : A ↝[  k ⌋-sym ] B}
    {f : B  A}
    {≡f : Extensionality c d 
           x  to-implication (inverse A↝B) x  f x}
    {x ext} 
  to-implication (inverse (with-other-inverse-ext? A↝B f ≡f ext)) x 
  f x
to-implication-inverse-with-other-inverse-ext? = λ where
  logical-equivalence  refl _
  bijection            refl _
  equivalence          refl _
  equivalenceᴱ         refl _

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

-- _⊎_ preserves all kinds of functions.

private

  ⊎-cong-inj :  {a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
                 {B₁ : Type b₁} {B₂ : Type 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-emb :  {a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
                 {B₁ : Type b₁} {B₂ : Type b₂} 
               Embedding A₁ A₂  Embedding B₁ B₂ 
               Embedding (A₁  B₁) (A₂  B₂)
  ⊎-cong-emb A₁↣A₂ B₁↣B₂ = record
    { to           = to′
    ; is-embedding = is-embedding′
    }
    where
    open Embedding

    to′ = ⊎-map (to A₁↣A₂) (to B₁↣B₂)

    is-embedding′ : Is-embedding to′
    is-embedding′ (inj₁ x) (inj₁ y) =
      _≃_.is-equivalence $
      Eq.with-other-function
        (inj₁ x  inj₁ y                        ↔⟨ inverse Bijection.≡↔inj₁≡inj₁ 
         x  y                                  ↝⟨ Eq.⟨ _ , is-embedding A₁↣A₂ _ _  
         to A₁↣A₂ x  to A₁↣A₂ y                ↔⟨ Bijection.≡↔inj₁≡inj₁ ⟩□
         inj₁ (to A₁↣A₂ x)  inj₁ (to A₁↣A₂ y)  )
        _
         eq 
           cong inj₁ (cong (to A₁↣A₂) (⊎.cancel-inj₁ eq))                 ≡⟨ cong-∘ _ _ _ 
           cong (inj₁  to A₁↣A₂) (⊎.cancel-inj₁ eq)                      ≡⟨ cong-∘ _ _ _ 
           cong (inj₁  to A₁↣A₂  [ id , const x ]) eq                   ≡⟨ sym $ trans-reflʳ _ 
           trans (cong (inj₁  to A₁↣A₂  [ id , const x ]) eq) (refl _)  ≡⟨ cong-respects-relevant-equality
                                                                               {f = inj₁  to A₁↣A₂  [ id , const x ]}
                                                                               (if_then true else false)
                                                                               [  _ _  refl _) ,  _ ()) ] 
           trans (refl _) (cong (⊎-map (to A₁↣A₂) (to B₁↣B₂)) eq)         ≡⟨ trans-reflˡ _ ⟩∎
           cong (⊎-map (to A₁↣A₂) (to B₁↣B₂)) eq                          )

    is-embedding′ (inj₂ x) (inj₂ y) =
      _≃_.is-equivalence $
      Eq.with-other-function
        (inj₂ x  inj₂ y                        ↔⟨ inverse Bijection.≡↔inj₂≡inj₂ 
         x  y                                  ↝⟨ Eq.⟨ _ , is-embedding B₁↣B₂ _ _  
         to B₁↣B₂ x  to B₁↣B₂ y                ↔⟨ Bijection.≡↔inj₂≡inj₂ ⟩□
         inj₂ (to B₁↣B₂ x)  inj₂ (to B₁↣B₂ y)  )
        _
         eq 
           cong inj₂ (cong (to B₁↣B₂) (⊎.cancel-inj₂ eq))                 ≡⟨ cong-∘ _ _ _ 
           cong (inj₂  to B₁↣B₂) (⊎.cancel-inj₂ eq)                      ≡⟨ cong-∘ _ _ _ 
           cong (inj₂  to B₁↣B₂  [ const x , id ]) eq                   ≡⟨ sym $ trans-reflʳ _ 
           trans (cong (inj₂  to B₁↣B₂  [ const x , id ]) eq) (refl _)  ≡⟨ cong-respects-relevant-equality
                                                                               {f = inj₂  to B₁↣B₂  [ const x , id ]}
                                                                               (if_then false else true)
                                                                               [  _ ()) ,  _ _  refl _) ] 
           trans (refl _) (cong (⊎-map (to A₁↣A₂) (to B₁↣B₂)) eq)         ≡⟨ trans-reflˡ _ ⟩∎
           cong (⊎-map (to A₁↣A₂) (to B₁↣B₂)) eq                          )

    is-embedding′ (inj₁ x) (inj₂ y) =
      _≃_.is-equivalence $
      Eq.with-other-function
        (inj₁ x  inj₂ y                        ↔⟨ inverse $ Bijection.⊥↔uninhabited ⊎.inj₁≢inj₂ 
         ⊥₀                                     ↔⟨ Bijection.⊥↔uninhabited ⊎.inj₁≢inj₂ ⟩□
         inj₁ (to A₁↣A₂ x)  inj₂ (to B₁↣B₂ y)  )
        _
        (⊥-elim  ⊎.inj₁≢inj₂)

    is-embedding′ (inj₂ x) (inj₁ y) =
      _≃_.is-equivalence $
      Eq.with-other-function
        (inj₂ x  inj₁ y                        ↔⟨ inverse $ Bijection.⊥↔uninhabited (⊎.inj₁≢inj₂  sym) 
         ⊥₀                                     ↔⟨ Bijection.⊥↔uninhabited (⊎.inj₁≢inj₂  sym) ⟩□
         inj₂ (to B₁↣B₂ x)  inj₁ (to A₁↣A₂ y)  )
        _
        (⊥-elim  ⊎.inj₁≢inj₂  sym)

  ⊎-cong-surj :  {a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
                  {B₁ : Type b₁} {B₂ : Type b₂} 
                A₁  A₂  B₁  B₂  A₁  B₁  A₂  B₂
  ⊎-cong-surj A₁↠A₂ B₁↠B₂ = record
    { logical-equivalence =
        _↠_.logical-equivalence A₁↠A₂
          L.⊎-cong
        _↠_.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 : Type a₁} {A₂ : Type a₂}
                 {B₁ : Type b₁} {B₂ : Type 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
        ]
    }

  ⊎-cong-≃ :
     {a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
      {B₁ : Type b₁} {B₂ : Type b₂} 
    A₁  A₂  B₁  B₂  (A₁  B₁)  (A₂  B₂)
  ⊎-cong-≃ A₁≃A₂ B₁≃B₂ =
    from-bijection $ ⊎-cong-bij (from-equivalence A₁≃A₂)
                                (from-equivalence B₁≃B₂)

  ⊎-cong-≃ᴱ :
     {a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
      {B₁ : Type b₁} {B₂ : Type b₂} 
    A₁ ≃ᴱ A₂  B₁ ≃ᴱ B₂  (A₁  B₁) ≃ᴱ (A₂  B₂)
  ⊎-cong-≃ᴱ f g =
    EEq.[≃]→≃ᴱ (EEq.[proofs] (⊎-cong-≃ (EEq.≃ᴱ→≃ f) (EEq.≃ᴱ→≃ g)))

infixr 1 _⊎-cong_

_⊎-cong_ :  {k a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
             {B₁ : Type b₁} {B₂ : Type b₂} 
           A₁ ↝[ k ] A₂  B₁ ↝[ k ] B₂  A₁  B₁ ↝[ k ] A₂  B₂
_⊎-cong_ {implication}         = ⊎-map
_⊎-cong_ {logical-equivalence} = L._⊎-cong_
_⊎-cong_ {injection}           = ⊎-cong-inj
_⊎-cong_ {embedding}           = ⊎-cong-emb
_⊎-cong_ {surjection}          = ⊎-cong-surj
_⊎-cong_ {bijection}           = ⊎-cong-bij
_⊎-cong_ {equivalence}         = ⊎-cong-≃
_⊎-cong_ {equivalenceᴱ}        = ⊎-cong-≃ᴱ

-- _⊎_ is commutative.

⊎-comm :  {a b} {A : Type a} {B : Type 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 : Type a} {B : Type b} {C : Type 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 : Type 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 : Type 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 : Type a}  A  A  A
⊎-idempotent = record
  { to   = [ id , id ]
  ; from = inj₁
  }

-- Lemmas that can be used to simplify binary sums where one of the
-- two type arguments is related to the empty type.

drop-⊥-right :
   {k a b} {A : Type a} {B : Type b} 
  B ↝[ k ] ⊥₀  A  B ↝[ k ] A
drop-⊥-right {A = A} {B} B↔⊥ =
  A  B  ↝⟨ id ⊎-cong B↔⊥ 
  A    ↔⟨ ⊎-right-identity ⟩□
  A      

drop-⊥-left :
   {k a b} {A : Type a} {B : Type b} 
  A ↝[ k ] ⊥₀  A  B ↝[ k ] B
drop-⊥-left {A = A} {B} A↔⊥ =
  A  B  ↔⟨ ⊎-comm 
  B  A  ↝⟨ drop-⊥-right A↔⊥ ⟩□
  B      

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

-- Σ preserves embeddings. (This definition is used in the proof of
-- _×-cong_.)

Σ-preserves-embeddings :
   {a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
    {B₁ : A₁  Type b₁} {B₂ : A₂  Type b₂}
  (A₁↣A₂ : Embedding A₁ A₂) 
  (∀ x  Embedding (B₁ x) (B₂ (Embedding.to A₁↣A₂ x))) 
  Embedding (Σ A₁ B₁) (Σ A₂ B₂)
Σ-preserves-embeddings {B₁ = B₁} {B₂} A₁↣A₂ B₁↣B₂ = record
  { to           = Σ-map (to A₁↣A₂) (to (B₁↣B₂ _))
  ; is-embedding = λ { (x₁ , x₂) (y₁ , y₂) 
      _≃_.is-equivalence $
      Eq.with-other-function
        ((x₁ , x₂)  (y₁ , y₂)                                   ↝⟨ inverse $ Eq.↔⇒≃ Bijection.Σ-≡,≡↔≡ 

         ( λ (eq : x₁  y₁)  subst B₁ eq x₂  y₂)              ↝⟨ Eq.Σ-preserves (Embedding.equivalence A₁↣A₂)  eq 

             subst B₁ eq x₂  y₂                                      ↝⟨ Embedding.equivalence (B₁↣B₂ y₁) 

             to (B₁↣B₂ y₁) (subst B₁ eq x₂)  to (B₁↣B₂ y₁) y₂        ↝⟨ ≡⇒↝ _ (cong (_≡ _) $ lemma₁ eq _ y₂) ⟩□

             subst B₂ (cong (to A₁↣A₂) eq) (to (B₁↣B₂ x₁) x₂) 
             to (B₁↣B₂ y₁) y₂                                         ) 

         ( λ (eq : to A₁↣A₂ x₁  to A₁↣A₂ y₁) 
            subst B₂ eq (to (B₁↣B₂ x₁) x₂)  to (B₁↣B₂ y₁) y₂)   ↝⟨ Eq.↔⇒≃ Bijection.Σ-≡,≡↔≡ ⟩□

         (to A₁↣A₂ x₁ , to (B₁↣B₂ x₁) x₂) 
         (to A₁↣A₂ y₁ , to (B₁↣B₂ y₁) y₂)                        )
        _
        (elim
           { {y = _ , y₂} eq 
               uncurry Σ-≡,≡→≡
                 (Σ-map (cong (to A₁↣A₂))
                        (_≃_.to (≡⇒↝ _ (cong (_≡ _) $ lemma₁ _ _ y₂)) 
                         cong (to (B₁↣B₂ _)))
                        (Σ-≡,≡←≡ eq)) 
               cong (Σ-map (to A₁↣A₂) (to (B₁↣B₂ _))) eq })
           _ 
             uncurry Σ-≡,≡→≡
               (Σ-map (cong (to A₁↣A₂))
                      (_≃_.to (≡⇒↝ _ (cong (_≡ _) $ lemma₁ _ _ _)) 
                       cong (to (B₁↣B₂ _)))
                      (Σ-≡,≡←≡ (refl _)))                                 ≡⟨ cong  eq  uncurry Σ-≡,≡→≡
                                                                                            (Σ-map _
                                                                                                   (_≃_.to (≡⇒↝ _ (cong (_≡ _) $ lemma₁ _ _ _)) 
                                                                                                    cong (to (B₁↣B₂ _)))
                                                                                                   eq))
                                                                                  Σ-≡,≡←≡-refl 
             Σ-≡,≡→≡
               (cong (to A₁↣A₂) (refl _))
               (_≃_.to (≡⇒↝ _ (cong (_≡ to (B₁↣B₂ _) _) $ lemma₁ _ _ _))
                  (cong (to (B₁↣B₂ _)) (subst-refl B₁ _)))                ≡⟨ Σ-≡,≡→≡-cong (cong-refl _) (lemma₂ _ _) 

             Σ-≡,≡→≡ (refl _) (subst-refl B₂ _)                           ≡⟨ Σ-≡,≡→≡-refl-subst-refl 

             refl _                                                       ≡⟨ sym $ cong-refl _ ⟩∎

             cong (Σ-map (to A₁↣A₂) (to (B₁↣B₂ _))) (refl _)              )) }
  }
  where
  open Embedding using (to)

  lemma₁ :  {x₁ y₁} (_ : x₁  y₁)  _
  lemma₁ = elim
     {x₁ y₁} eq  (x₂ : B₁ x₁) (y₂ : B₁ y₁) 
       to (B₁↣B₂ y₁) (subst B₁ eq x₂) 
       subst B₂ (cong (to A₁↣A₂) eq) (to (B₁↣B₂ x₁) x₂))
     z₁ x₂ y₂ 
       to (B₁↣B₂ z₁) (subst B₁ (refl z₁) x₂)                    ≡⟨ cong (to (B₁↣B₂ z₁)) $ subst-refl _ _ 
       to (B₁↣B₂ z₁) x₂                                         ≡⟨ sym $ subst-refl _ _ 
       subst B₂ (refl (to A₁↣A₂ z₁)) (to (B₁↣B₂ z₁) x₂)         ≡⟨ cong  eq  subst B₂ eq _) (sym $ cong-refl _) ⟩∎
       subst B₂ (cong (to A₁↣A₂) (refl z₁)) (to (B₁↣B₂ z₁) x₂)  )

  lemma₂ = λ x y 
    let eq₁ = cong (flip (subst B₂) _) (sym (cong-refl _))
        eq₂ = cong (to (B₁↣B₂ x)) (subst-refl B₁ y)
    in
    trans eq₁ (_≃_.to (≡⇒↝ _ (cong (_≡ _) $ lemma₁ (refl x) y y)) eq₂)   ≡⟨ cong  eq  trans eq₁ (_≃_.to (≡⇒↝ _ (cong (_≡ _) (eq y y))) eq₂)) $
                                                                              elim-refl  {x₁ y₁} eq  (x₂ : B₁ x₁) (y₂ : B₁ y₁) 
                                                                                           to (B₁↣B₂ y₁) (subst B₁ eq x₂) 
                                                                                           subst B₂ (cong (to A₁↣A₂) eq) (to (B₁↣B₂ x₁) x₂))
                                                                                        _ 
    trans eq₁ (_≃_.to (≡⇒↝ _ $ cong (_≡ _) $
                         trans eq₂ (trans (sym $ subst-refl B₂ _) eq₁))
                 eq₂)                                                    ≡⟨ cong (trans _) $ sym $ subst-in-terms-of-≡⇒↝ equivalence _ _ _ 

    trans eq₁ (subst (_≡ _)
                 (trans eq₂ (trans (sym $ subst-refl B₂ _) eq₁))
                 eq₂)                                                    ≡⟨ cong  eq  trans eq₁ (subst (_≡ _) eq eq₂)) $
                                                                              sym $ sym-sym (trans eq₂ (trans (sym $ subst-refl B₂ _) eq₁)) 
    trans eq₁ (subst (_≡ _)
                 (sym $ sym $
                    trans eq₂ (trans (sym $ subst-refl B₂ _) eq₁))
                 eq₂)                                                    ≡⟨ cong (trans _) $ subst-trans _ 

    trans eq₁ (trans
                 (sym $ trans eq₂ (trans (sym $ subst-refl B₂ _) eq₁))
                 eq₂)                                                    ≡⟨ cong  eq  trans eq₁ (trans eq eq₂)) $
                                                                              sym-trans eq₂ (trans (sym $ subst-refl B₂ _) eq₁) 
    trans eq₁ (trans (trans (sym $ trans (sym $ subst-refl B₂ _) eq₁)
                            (sym eq₂))
                     eq₂)                                                ≡⟨ cong (trans _) $ trans-[trans-sym]- _ _ 

    trans eq₁ (sym $ trans (sym $ subst-refl B₂ _) eq₁)                  ≡⟨ cong (trans _) $ sym-trans _ _ 

    trans eq₁ (trans (sym eq₁) (sym $ sym $ subst-refl B₂ _))            ≡⟨ trans--[trans-sym] _ _ 

    sym $ sym $ subst-refl B₂ _                                          ≡⟨ sym-sym _ ⟩∎

    subst-refl B₂ _                                                      

-- _×_ preserves all kinds of functions.

private

  ×-cong-inj :  {a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
                 {B₁ : Type b₁} {B₂ : Type 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 : Type a₁} {A₂ : Type a₂}
                  {B₁ : Type b₁} {B₂ : Type b₂} 
                A₁  A₂  B₁  B₂  A₁ × B₁  A₂ × B₂
  ×-cong-surj A₁↠A₂ B₁↠B₂ = record
    { logical-equivalence =
        _↠_.logical-equivalence A₁↠A₂
          L.×-cong
        _↠_.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 : Type a₁} {A₂ : Type a₂}
                 {B₁ : Type b₁} {B₂ : Type 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)
    }

  ×-cong-≃ :
     {a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
      {B₁ : Type b₁} {B₂ : Type b₂} 
    A₁  A₂  B₁  B₂  (A₁ × B₁)  (A₂ × B₂)
  ×-cong-≃ A₁≃A₂ B₁≃B₂ =
    from-bijection $ ×-cong-bij (from-equivalence A₁≃A₂)
                                (from-equivalence B₁≃B₂)

  ×-cong-≃ᴱ :
     {a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
      {B₁ : Type b₁} {B₂ : Type b₂} 
    A₁ ≃ᴱ A₂  B₁ ≃ᴱ B₂  (A₁ × B₁) ≃ᴱ (A₂ × B₂)
  ×-cong-≃ᴱ f g =
    EEq.[≃]→≃ᴱ (EEq.[proofs] (×-cong-≃ (EEq.≃ᴱ→≃ f) (EEq.≃ᴱ→≃ g)))

infixr 2 _×-cong_

_×-cong_ :  {k a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
             {B₁ : Type b₁} {B₂ : Type b₂} 
           A₁ ↝[ k ] A₂  B₁ ↝[ k ] B₂  A₁ × B₁ ↝[ k ] A₂ × B₂
_×-cong_ {implication}         = λ f g  Σ-map f g
_×-cong_ {logical-equivalence} = L._×-cong_
_×-cong_ {injection}           = ×-cong-inj
_×-cong_ {embedding}           = λ A₁↣A₂ B₁↣B₂ 
                                   Σ-preserves-embeddings
                                     A₁↣A₂  _  B₁↣B₂)
_×-cong_ {surjection}          = ×-cong-surj
_×-cong_ {bijection}           = ×-cong-bij
_×-cong_ {equivalence}         = ×-cong-≃
_×-cong_ {equivalenceᴱ}        = ×-cong-≃ᴱ

-- The function to-implication is homomorphic with respect to
-- _×-cong_/Σ-map.

to-implication-×-cong :
   k {a b c d} {A : Type a} {B : Type b} {C : Type c} {D : Type d}
    {A↝B : A ↝[ k ] B} {C↝D : C ↝[ k ] D} 
  to-implication (A↝B ×-cong C↝D) 
  Σ-map (to-implication A↝B) (to-implication C↝D)
to-implication-×-cong implication         = refl _
to-implication-×-cong logical-equivalence = refl _
to-implication-×-cong injection           = refl _
to-implication-×-cong embedding           = refl _
to-implication-×-cong surjection          = refl _
to-implication-×-cong bijection           = refl _
to-implication-×-cong equivalence         = refl _
to-implication-×-cong equivalenceᴱ        = refl _

-- _×_ is commutative.

×-comm :  {a b} {A : Type a} {B : Type 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.

open Bijection public using (Σ-assoc)

-- _×_ is associative.

×-assoc :  {a b c} {A : Type a} {B : Type b} {C : Type c} 
          A × (B × C)  (A × B) × C
×-assoc = Σ-assoc

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

Σ-left-identity :  {a} {A :   Type 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 : Type a}   × A  A
×-left-identity = Σ-left-identity

×-right-identity :  {a} {A : Type 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}  Type 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 : Type a}   { = ℓ₁} × A   { = ℓ₂}
×-left-zero = Σ-left-zero

×-right-zero :  {a ℓ₁ ℓ₂} {A : Type 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.

Σ-cong :  {k₁ k₂ a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
           {B₁ : A₁  Type b₁} {B₂ : A₂  Type 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} {implication}         = λ A₁≃A₂ B₁→B₂ 
                                               Σ-map (from-isomorphism A₁≃A₂) (B₁→B₂ _)
Σ-cong {bijection}   {implication}         = λ A₁↔A₂ B₁→B₂ 
                                               Σ-map (from-isomorphism A₁↔A₂) (B₁→B₂ _)
Σ-cong {equivalence} {logical-equivalence} = Surjection.Σ-cong-⇔        from-isomorphism
Σ-cong {bijection}   {logical-equivalence} = Surjection.Σ-cong-⇔        from-isomorphism
Σ-cong {equivalence} {injection}           = Eq.∃-preserves-injections
Σ-cong {bijection}   {injection}           = Eq.∃-preserves-injections  from-isomorphism
Σ-cong {equivalence} {embedding}           = Σ-preserves-embeddings     from-isomorphism
Σ-cong {bijection}   {embedding}           = Σ-preserves-embeddings     from-isomorphism
Σ-cong {equivalence} {surjection}          = Surjection.Σ-cong          from-isomorphism
Σ-cong {bijection}   {surjection}          = Surjection.Σ-cong          from-isomorphism
Σ-cong {equivalence} {bijection}           = Eq.∃-preserves-bijections
Σ-cong {bijection}   {bijection}           = Eq.∃-preserves-bijections  from-isomorphism
Σ-cong {equivalence} {equivalence}         = Eq.Σ-preserves
Σ-cong {bijection}   {equivalence}         = Eq.Σ-preserves             from-isomorphism
Σ-cong {equivalence} {equivalenceᴱ}
       {B₂ = B₂}                           = λ f g 
  EEq.[≃]→≃ᴱ
    {to   = λ (x , y)  _≃_.to f x , _≃ᴱ_.to (g x) y}
    {from = λ (x , y) 
                _≃_.from f x
              , _≃ᴱ_.from (g (_≃_.from f x))
                   (subst B₂ (sym (_≃_.right-inverse-of f x)) y)}
    (EEq.[proofs]
       (Eq.Σ-preserves f (EEq.≃ᴱ→≃  g)))
Σ-cong {bijection}   {equivalenceᴱ}
       {B₂ = B₂}                           = λ f g 
  EEq.[≃]→≃ᴱ
    {to   = λ (x , y)  _↔_.to f x , _≃ᴱ_.to (g x) y}
    {from = λ (x , y) 
                _↔_.from f x
              , _≃ᴱ_.from (g (_↔_.from f x))
                   (subst B₂
                      (sym (_≃_.right-inverse-of (Eq.↔⇒≃ f) x))
                      y)}
    (EEq.[proofs]
       (Eq.Σ-preserves (from-isomorphism f)
          (EEq.≃ᴱ→≃  g)))

-- A variant of Σ-cong.

Σ-cong-contra :
   {k₁ k₂ a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
    {B₁ : A₁  Type b₁} {B₂ : A₂  Type b₂} 
  (A₂↔A₁ : A₂ ↔[ k₁ ] A₁) 
  (∀ x  B₁ (to-implication A₂↔A₁ x) ↝[ k₂ ] B₂ x) 
  Σ A₁ B₁ ↝[ k₂ ] Σ A₂ B₂
Σ-cong-contra {k₂ = logical-equivalence} A₂↔A₁ B₁⇔B₂ =
  inverse $ Σ-cong A₂↔A₁ (inverse  B₁⇔B₂)
Σ-cong-contra {k₂ = bijection} A₂↔A₁ B₁↔B₂ =
  inverse $ Σ-cong A₂↔A₁ (inverse  B₁↔B₂)
Σ-cong-contra {k₂ = equivalence} A₂↔A₁ B₁≃B₂ =
  inverse $ Σ-cong A₂↔A₁ (inverse  B₁≃B₂)
Σ-cong-contra {k₂ = equivalenceᴱ} A₂↔A₁ B₁≃ᴱB₂ =
  inverse $ Σ-cong A₂↔A₁ (inverse  B₁≃ᴱB₂)
Σ-cong-contra {k₁} {k₂} {A₁ = A₁} {A₂} {B₁} {B₂} A₂↔A₁ B₁↝B₂ =
  Σ-cong A₁↔A₂ B₁↝B₂′
  where
  A₁↔A₂ : A₁  A₂
  A₁↔A₂ = inverse $ from-isomorphism A₂↔A₁

  B₁↝B₂′ :  x  B₁ x ↝[ k₂ ] B₂ (_↔_.to A₁↔A₂ x)
  B₁↝B₂′ x =
    B₁ x                                        ↝⟨ ≡⇒↝ _ $ cong B₁ $ sym $ _↔_.left-inverse-of A₁↔A₂ _ 
    B₁ (_↔_.from A₁↔A₂ (_↔_.to A₁↔A₂ x))        ↝⟨ ≡⇒↝ _ $ cong  f  B₁ (f (_↔_.to A₁↔A₂ x))) $ sym $
                                                     to-implication∘from-isomorphism k₁ bijection 
    B₁ (to-implication A₂↔A₁ (_↔_.to A₁↔A₂ x))  ↝⟨ B₁↝B₂ (_↔_.to A₁↔A₂ x) ⟩□
    B₂ (_↔_.to A₁↔A₂ x)                         

-- Variants of special cases of Σ-cong-contra.

Σ-cong-contra-→ :
   {a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
    {B₁ : A₁  Type b₁} {B₂ : A₂  Type b₂} 
  (A₂↠A₁ : A₂  A₁) 
  (∀ x  B₁ (_↠_.to A₂↠A₁ x)  B₂ x) 
  Σ A₁ B₁  Σ A₂ B₂
Σ-cong-contra-→ {B₁ = B₁} A₂↠A₁ B₁→B₂ =
  Σ-map (_↠_.from A₂↠A₁)
        (B₁→B₂ _  subst B₁ (sym $ _↠_.right-inverse-of A₂↠A₁ _))

Σ-cong-contra-⇔ :
   {a₁ a₂ b₁ b₂} {A₁ : Type a₁} {A₂ : Type a₂}
    {B₁ : A₁  Type b₁} {B₂ : A₂  Type b₂} 
  (A₂↠A₁ : A₂  A₁) 
  (∀ x  B₁ (_↠_.to A₂↠A₁ x)  B₂ x) 
  Σ A₁ B₁  Σ A₂ B₂
Σ-cong-contra-⇔ A₂↠A₁ B₁⇔B₂ =
  inverse $
  Surjection.Σ-cong-⇔ A₂↠A₁ (inverse  B₁⇔B₂)

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

  ∃-cong-surj :  {a b₁ b₂}
                  {A : Type a} {B₁ : A  Type b₁} {B₂ : A  Type b₂} 
                (∀ x  B₁ x  B₂ x)   B₁   B₂
  ∃-cong-surj B₁↠B₂ = record
    { logical-equivalence = L.∃-cong (_↠_.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 : Type a} {B₁ : A  Type b₁} {B₂ : A  Type 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-≃ᴱ :
     {a b₁ b₂}
      {A : Type a} {B₁ : A  Type b₁} {B₂ : A  Type b₂} 
    (∀ x  B₁ x ≃ᴱ B₂ x)   B₁ ≃ᴱ  B₂
  ∃-cong-≃ᴱ f = EEq.[≃]→≃ᴱ (EEq.[proofs] (Eq.∃-cong (EEq.≃ᴱ→≃  f)))

∃-cong :  {k a b₁ b₂}
           {A : Type a} {B₁ : A  Type b₁} {B₂ : A  Type b₂} 
         (∀ x  B₁ x ↝[ k ] B₂ x)   B₁ ↝[ k ]  B₂
∃-cong {implication}         = ∃-cong-impl
∃-cong {logical-equivalence} = L.∃-cong
∃-cong {injection}           = Σ-cong Bijection.id
∃-cong {embedding}           = Σ-preserves-embeddings Emb.id
∃-cong {surjection}          = ∃-cong-surj
∃-cong {bijection}           = ∃-cong-bij
∃-cong {equivalence}         = Eq.∃-cong
∃-cong {equivalenceᴱ}        = ∃-cong-≃ᴱ

private

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

  ×-cong₂ :  {k a b₁ b₂}
              {A : Type a} {B₁ : Type b₁} {B₂ : Type 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 : Type a₁} {A₂ : Type a₂} {B : Type 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) related to the unit type.

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

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

drop-⊤-left-Σ :  {a b} {A : Type a} {B : A  Type 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 : Type a} {B : A  Type b} {C : Σ A B  Type 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 : Type a} {B : Type b} {C : A  B  Type 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 : Type a} {B : Type b} {C : A  B  Type c} 
  ((x : A  B)  C x)
    ↝[ a  b  c ]
  ((x : A)  C (inj₁ x)) × ((y : B)  C (inj₂ y))
Π⊎↔Π×Π =
  generalise-ext?
    (_↠_.logical-equivalence Π⊎↠Π×Π)
     ext 
         refl
       ,  _  apply-ext ext [ refl  _ , refl  _ ]))

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

∃-⊎-distrib-left :
   {a b c} {A : Type a} {B : A  Type b} {C : A  Type 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 : Type a} {B : Type b} {C : A  B  Type 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 : Type a} {B : Type b} {C : A  B  Type 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 : Type a} (B : A  Type b) (x : A) 
          B x   λ y  B y × y  x
∃-intro B x = _≃_.bijection $ Eq.↔→≃
   b  x , b , refl _)
   (y , b , y≡x)  subst B y≡x b)
   (y , b , y≡x) 
     sym $
     Σ-≡,≡→≡
       y≡x
       (subst  y  B y × y  x) y≡x (b , y≡x)  ≡⟨ push-subst-, _ _ 
        subst B y≡x b , subst (_≡ x) y≡x y≡x     ≡⟨ cong (_ ,_) subst-trans-sym 
        subst B y≡x b , trans (sym y≡x) y≡x      ≡⟨ cong (_ ,_) $ trans-symˡ _ ⟩∎
        subst B y≡x b , refl x                   ))
  (subst-refl B)

-- A variant of ∃-intro.

other-∃-intro :
   {a b} {A : Type a} (B : A  Type b) (x : A) 
  B x   λ y  B y × x  y
other-∃-intro B x = Eq.↔→≃
   b  x , b , refl _)
   (y , b , x≡y)  subst B (sym x≡y) b)
   (y , b , x≡y) 
     Σ-≡,≡→≡
       x≡y
       (subst  y  B y × x  y) x≡y (subst B (sym x≡y) b , refl x)   ≡⟨ push-subst-, _ _ 
        subst B x≡y (subst B (sym x≡y) b) , subst (x ≡_) x≡y (refl x)  ≡⟨ cong₂ _,_
                                                                            (subst-subst-sym _ _ _)
                                                                            (trans (sym trans-subst) $
                                                                             trans-reflˡ _) ⟩∎
        b , x≡y                                                        ))
   b 
     subst B (sym (refl _)) b  ≡⟨ cong (flip (subst B) _) sym-refl 
     subst B (refl _) b        ≡⟨ subst-refl _ _ ⟩∎
     b                         )

-- Another variant of ∃-intro.

∃-introduction :
   {a b} {A : Type a} {x : A} (B : (y : A)  x  y  Type 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 _ 
                                                                  _⇔_.to contractible⇔↔⊤ $
                                                                  ⇒≡ 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-≡,≡↔≡.
--
-- This property used to be defined in terms of Σ-≡,≡↔≡, but was
-- changed in order to make it compute in a different way.

≡×≡↔≡ :  {a b} {A : Type a} {B : Type b} {p₁ p₂ : A × B} 
        (proj₁ p₁  proj₁ p₂ × proj₂ p₁  proj₂ p₂)  (p₁  p₂)
≡×≡↔≡ {B = B} {p₁} {p₂} = record
  { surjection = record
    { logical-equivalence = record
      { to   = uncurry (cong₂ _,_)
      ; from = λ eq  cong proj₁ eq , cong proj₂ eq
      }
    ; right-inverse-of = λ eq 
        cong₂ _,_ (cong proj₁ eq) (cong proj₂ eq)  ≡⟨ cong₂-cong-cong _ _ _,_ 
        cong  p  proj₁ p , proj₂ p) eq          ≡⟨⟩
        cong id eq                                 ≡⟨ sym $ cong-id _ ⟩∎
        eq                                         
    }
  ; left-inverse-of = λ { (eq₁ , eq₂) 
        cong proj₁ (cong₂ _,_ eq₁ eq₂) , cong proj₂ (cong₂ _,_ eq₁ eq₂)  ≡⟨ cong₂ _,_ (cong-proj₁-cong₂-, eq₁ eq₂) (cong-proj₂-cong₂-, eq₁ eq₂) ⟩∎
        eq₁ , eq₂                                                        
      }
  }

-- 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 : Type a} {B : A  Type 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  _  inverse $ _⇔_.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 : Type x} {Y : Type y} 
  Contractible (X × Y) ↝[ x  y  x  y ]
  (Contractible X × Contractible Y)
Contractible-commutes-with-× {x = x} {y} =
  generalise-ext?-prop
    (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′)   } }
       })
    Contractible-propositional
     ext  ×-closure 1 (Contractible-propositional
                            (lower-extensionality y y ext))
                         (Contractible-propositional
                            (lower-extensionality x x ext)))
  where
  lemma :  {x y} {X : Type x} {Y : Type 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 relating equality of certain kinds of functions to
-- pointwise equality of the underlying functions

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

≃-to-≡↔≡ :
   {a b} 
  Extensionality (a  b) (a  b) 
  {A : Type a} {B : Type 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 (Is-equivalence-propositional ext) 
  (_≃_.to p , _≃_.is-equivalence p)  (_≃_.to q , _≃_.is-equivalence q)  ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ Eq.≃-as-Σ) ⟩□
  p  q                                                                  

-- A variant of the previous result for which both directions compute
-- in certain ways.

≃-to-≡≃≡ :
   {a b} 
  Extensionality (a  b) (a  b) 
  Extensionality a b 
  {A : Type a} {B : Type b} {p q : A  B} 
  (∀ x  _≃_.to p x  _≃_.to q x)  (p  q)
≃-to-≡≃≡ ext₁ ext₂ {p = p} {q = q} =
  Eq.↔→≃
    (Eq.lift-equality ext₁  apply-ext ext₂)
    (flip $ cong  flip _≃_.to)
    (elim¹
        p≡q 
          Eq.lift-equality ext₁
            (apply-ext ext₂
                x  cong  eq  _≃_.to eq x) p≡q)) 
          p≡q)
       (Eq.lift-equality ext₁
          (apply-ext ext₂
              x  cong  eq  _≃_.to eq x) (refl _)))  ≡⟨ (cong (Eq.lift-equality ext₁) $
                                                              cong (apply-ext ext₂) $
                                                              apply-ext ext₂ λ _ 
                                                              cong-refl _) 
        Eq.lift-equality ext₁
          (apply-ext ext₂  _  refl _))                 ≡⟨ cong (Eq.lift-equality ext₁) $
                                                             ext-refl ext₂ 

        Eq.lift-equality ext₁ (refl _)                    ≡⟨ Eq.lift-equality-refl ext₁ 

        cong Eq.⟨ _≃_.to p ,_⟩ _                          ≡⟨ cong (cong Eq.⟨ _≃_.to p ,_⟩) $
                                                             mono₁ 1 (Is-equivalence-propositional ext₁) _ _ 

        cong Eq.⟨ _≃_.to p ,_⟩ (refl _)                   ≡⟨ cong-refl _ ⟩∎

        refl _                                            ))
     p≡q  apply-ext ext₂ λ x 
       cong  eq  _≃_.to eq x)
         (Eq.lift-equality ext₁
            (apply-ext ext₂ p≡q))                  ≡⟨ elim¹
                                                         {g} p≡g 
                                                           (eq : Is-equivalence g) 
                                                           cong  eq  _≃_.to eq x)
                                                             (Eq.lift-equality ext₁ {q = Eq.⟨ _ , eq } p≡g) 
                                                           ext⁻¹ p≡g x)
                                                         eq 
           cong  eq  _≃_.to eq x)
             (Eq.lift-equality ext₁ (refl _))              ≡⟨ cong (cong _) $ Eq.lift-equality-refl ext₁ 

           cong  eq  _≃_.to eq x)
             (cong Eq.⟨ _≃_.to p ,_⟩ _)                    ≡⟨ cong-∘ _ _ _ 

           cong (const (_≃_.to p x)) _                     ≡⟨ cong-const _ 

           refl _                                          ≡⟨ sym $ cong-refl _ ⟩∎

           ext⁻¹ (refl _) x                                )
                                                        (apply-ext ext₂ p≡q)
                                                        _ 

       ext⁻¹ (apply-ext ext₂ p≡q) x                ≡⟨ cong (_$ x) $
                                                      _≃_.left-inverse-of (Eq.extensionality-isomorphism ext₂) _ ⟩∎
       p≡q x                                       )

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

≃-from-≡↔≡ :
   {a b} 
  Extensionality (a  b) (a  b) 
  {A : Type a} {B : Type 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 functions (assuming
-- extensionality).

↔-to-≡↔≡ :
   {a b} 
  Extensionality (a  b) (a  b) 
  {A : Type a} {B : Type 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 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 : Type a} {B : Type 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                                                  

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

Embedding-to-≡↔≡ :
   {a b} 
  Extensionality (a  b) (a  b) 
  {A : Type a} {B : Type b} {p q : Embedding A B} 
  (∀ x  Embedding.to p x  Embedding.to q x)  p  q
Embedding-to-≡↔≡ {a} {b} ext {p = p} {q} =
  (∀ x  Embedding.to p x  Embedding.to q x)    ↔⟨ Eq.extensionality-isomorphism (lower-extensionality b a ext) 

  Embedding.to p  Embedding.to q                ↝⟨ ignore-propositional-component (Emb.Is-embedding-propositional ext) 

  (Embedding.to p , Embedding.is-embedding p) 
  (Embedding.to q , Embedding.is-embedding q)    ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ Emb.Embedding-as-Σ) ⟩□

  p  q                                          

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

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

×-⊎-distrib-left :  {a b c} {A : Type a} {B : Type b} {C : Type c} 
                   A × (B  C)  (A × B)  (A × C)
×-⊎-distrib-left = ∃-⊎-distrib-left

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

×-⊎-distrib-right :  {a b c} {A : Type a} {B : Type b} {C : Type c} 
                    (A  B) × C  (A × C)  (B × C)
×-⊎-distrib-right = ∃-⊎-distrib-right

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

-- The non-dependent function space preserves non-dependent functions
-- (contravariantly for the domain).

→-cong-→ :  {a b c d}
             {A : Type a} {B : Type b} {C : Type c} {D : Type d} 
           (B  A)  (C  D)  (A  C)  (B  D)
→-cong-→ B→A C→D = (C→D ∘_)  (_∘ B→A)

-- The non-dependent function space preserves split surjections
-- (assuming extensionality).

→-cong-↠ :  {a b c d}  Extensionality b d 
           {A : Type a} {B : Type b} {C : Type c} {D : Type d} 
           A  B  C  D  (A  C)  (B  D)
→-cong-↠ {a} {b} {c} {d} ext A↠B C↠D = record
  { logical-equivalence = logical-equiv
  ; right-inverse-of    = right-inv
  }
  where
  open _↠_

  logical-equiv = L.→-cong (_↠_.logical-equivalence A↠B)
                           (_↠_.logical-equivalence C↠D)

  abstract
    right-inv :
       f  _⇔_.to logical-equiv (_⇔_.from logical-equiv f)  f
    right-inv f = apply-ext 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                                          

private

  -- Lemmas used in the implementation of →-cong.

  →-cong-↔ :  {a b c d}
               {A : Type a} {B : Type b} {C : Type c} {D : Type d} 
             Extensionality (a  b) (c  d) 
             A  B  C  D  (A  C)  (B  D)
  →-cong-↔ {a} {b} {c} {d} ext A↔B C↔D = record
    { surjection      = surj
    ; left-inverse-of = left-inv
    }
    where
    open _↔_

    surj = →-cong-↠ (lower-extensionality a c ext)
                    (_↔_.surjection A↔B)
                    (_↔_.surjection C↔D)

    abstract
      left-inv :
         f  _↠_.from surj (_↠_.to surj f)  f
      left-inv f = apply-ext (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                                          

  →-cong-≃ :
     {a b c d}
      {A : Type a} {B : Type b} {C : Type c} {D : Type d} 
    Extensionality (a  b) (c  d) 
    A  B  C  D  (A  C)  (B  D)
  →-cong-≃ ext A≃B C≃D = record
    { to             = to
    ; is-equivalence = from , proofs
    }
    where
    A→C≃B→D =
      Eq.↔⇒≃ (→-cong-↔ ext (_≃_.bijection A≃B) (_≃_.bijection C≃D))

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

    abstract

      proofs : HA.Proofs to from
      proofs = proj₂ (_≃_.is-equivalence A→C≃B→D)

  →-cong-≃ᴱ :
     {a b c d}
      {A : Type a} {B : Type b} {C : Type c} {D : Type d} 
    Erased (Extensionality (a  b) (c  d)) 
    A ≃ᴱ B  C ≃ᴱ D  (A  C) ≃ᴱ (B  D)
  →-cong-≃ᴱ E.[ ext ] f g =
    EEq.[≃]→≃ᴱ (EEq.[proofs] (→-cong-≃ ext (EEq.≃ᴱ→≃ f) (EEq.≃ᴱ→≃ g)))

-- The non-dependent function space preserves symmetric kinds of
-- functions (in some cases assuming extensionality).

→-cong :  {k a b c d} 
         Extensionality?  k ⌋-sym (a  b) (c  d) 
         {A : Type a} {B : Type b} {C : Type c} {D : Type d} 
         A ↝[  k ⌋-sym ] B  C ↝[  k ⌋-sym ] D 
         (A  C) ↝[  k ⌋-sym ] (B  D)
→-cong {logical-equivalence} _   = L.→-cong
→-cong {bijection}           ext = →-cong-↔  ext
→-cong {equivalence}         ext = →-cong-≃  ext
→-cong {equivalenceᴱ}        ext = →-cong-≃ᴱ ext

-- A variant of →-cong.

→-cong₁ :
   {k₁ k₂ a b c} 
  Extensionality? k₂ (a  b) c 
  {A : Type a} {B : Type b} {C : Type c} 
  A ↔[ k₁ ] B  (A  C) ↝[ k₂ ] (B  C)
→-cong₁ ext hyp = generalise-ext?-sym
   ext  →-cong ext (from-bijection (from-isomorphism hyp)) id)
  ext

private

  -- Lemmas used in the implementation of ∀-cong.

  ∀-cong-→ :
     {a b₁ b₂} {A : Type a} {B₁ : A  Type b₁} {B₂ : A  Type b₂} 
    (∀ x  B₁ x  B₂ x) 
    ((x : A)  B₁ x)  ((x : A)  B₂ x)
  ∀-cong-→ B₁→B₂ = B₁→B₂ _ ⊚_

  ∀-cong-bij :
     {a b₁ b₂} 
    Extensionality a (b₁  b₂) 
    {A : Type a} {B₁ : A  Type b₁} {B₂ : A  Type b₂} 
    (∀ x  B₁ x  B₂ x) 
    ((x : A)  B₁ x)  ((x : A)  B₂ x)
  ∀-cong-bij {b₂ = b₂} ext B₁↔B₂ = record
    { surjection      = surj
    ; left-inverse-of = left-inverse-of
    }
    where
    surj = ∀-cong-↠ ext (_↔_.surjection  B₁↔B₂)

    abstract
      left-inverse-of :  f  _↠_.from surj (_↠_.to surj f)  f
      left-inverse-of = λ f 
        apply-ext (lower-extensionality lzero b₂ ext) λ x 
          _↔_.from (B₁↔B₂ x) (_↔_.to (B₁↔B₂ x) (f x))  ≡⟨ _↔_.left-inverse-of (B₁↔B₂ x) (f x) ⟩∎
          f x                                          

  ∀-cong-eq :
     {a b₁ b₂} 
    Extensionality a (b₁  b₂) 
    {A : Type a} {B₁ : A  Type b₁} {B₂ : A  Type b₂} 
    (∀ x  B₁ x  B₂ x) 
    ((x : A)  B₁ x)  ((x : A)  B₂ x)
  ∀-cong-eq ext = Eq.↔⇒≃  ∀-cong-bij ext  (_≃_.bijection ⊚_)

  ∀-cong-eqᴱ :
     {a b₁ b₂} 
    Erased (Extensionality a (b₁  b₂)) 
    {A : Type a} {B₁ : A  Type b₁} {B₂ : A  Type b₂} 
    (∀ x  B₁ x ≃ᴱ B₂ x) 
    ((x : A)  B₁ x) ≃ᴱ ((x : A)  B₂ x)
  ∀-cong-eqᴱ E.[ ext ] f =
    EEq.[≃]→≃ᴱ (EEq.[proofs] (∀-cong-eq ext (EEq.≃ᴱ→≃  f)))

  ∀-cong-inj :
     {a b₁ b₂} 
    Extensionality a (b₁  b₂) 
    {A : Type a} {B₁ : A  Type b₁} {B₂ : A  Type b₂} 
    (∀ x  B₁ x  B₂ x) 
    ((x : A)  B₁ x)  ((x : A)  B₂ x)
  ∀-cong-inj {b₁ = b₁} {b₂} ext B₁↣B₂ = record
    { to        = to
    ; injective = injective
    }
    where
    to = ∀-cong-→ (_↣_.to  B₁↣B₂)

    abstract
      injective : Injective to
      injective {x = f} {y = g} =
         x  _↣_.to (B₁↣B₂ x) (f x))   x  _↣_.to (B₁↣B₂ x) (g x))  ↔⟨ inverse $ Eq.extensionality-isomorphism
                                                                                        (lower-extensionality lzero b₁ ext) 
        (∀ x  _↣_.to (B₁↣B₂ x) (f x)  _↣_.to (B₁↣B₂ x) (g x))          ↝⟨ ∀-cong-→  x  _↣_.injective (B₁↣B₂ x)) 
        (∀ x  f x  g x)                                                ↔⟨ Eq.extensionality-isomorphism
                                                                              (lower-extensionality lzero b₂ ext) ⟩□
        f  g                                                            

  ∀-cong-emb :
     {a b₁ b₂} 
    Extensionality a (b₁  b₂) 
    {A : Type a} {B₁ : A  Type b₁} {B₂ : A  Type b₂} 
    (∀ x  Embedding (B₁ x) (B₂ x)) 
    Embedding ((x : A)  B₁ x) ((x : A)  B₂ x)
  ∀-cong-emb {b₁ = b₁} {b₂} ext B₁↣B₂ = record
    { to           = to
    ; is-embedding = is-embedding
    }
    where
    to = ∀-cong-→ (Embedding.to  B₁↣B₂)

    ext₂ = lower-extensionality lzero b₁ ext

    abstract
      is-embedding : Is-embedding to
      is-embedding f g = _≃_.is-equivalence $
        Eq.with-other-function
          (f  g                                   ↝⟨ inverse $ Eq.extensionality-isomorphism
                                                                  (lower-extensionality lzero b₂ ext) 
           (∀ x  f x  g x)                       ↝⟨ ∀-cong-eq ext  x 
                                                        Eq.⟨ _ , Embedding.is-embedding (B₁↣B₂ x) (f x) (g x) ) 
           (∀ x  Embedding.to (B₁↣B₂ x) (f x) 
                  Embedding.to (B₁↣B₂ x) (g x))    ↝⟨ Eq.extensionality-isomorphism ext₂ ⟩□

            x  Embedding.to (B₁↣B₂ x) (f x)) 
            x  Embedding.to (B₁↣B₂ x) (g x))    )
          _
           f≡g 
             apply-ext ext₂
                x  cong (Embedding.to (B₁↣B₂ x)) (ext⁻¹ f≡g x))        ≡⟨⟩

             apply-ext ext₂
                x  cong (Embedding.to (B₁↣B₂ x)) (cong (_$ x) f≡g))    ≡⟨ cong (apply-ext ext₂) (apply-ext ext₂ λ _ 
                                                                               cong-∘ _ _ _) 
             apply-ext ext₂
                x  cong  h  Embedding.to (B₁↣B₂ x) (h x)) f≡g)      ≡⟨ cong (apply-ext ext₂) (apply-ext ext₂ λ _  sym $
                                                                               cong-∘ _ _ _) 
             apply-ext ext₂
                x  cong (_$ x)
                        (cong  h x  Embedding.to (B₁↣B₂ x) (h x))
                           f≡g))                                          ≡⟨⟩

             apply-ext ext₂
               (ext⁻¹ (cong  h x  Embedding.to (B₁↣B₂ x) (h x)) f≡g))  ≡⟨ _≃_.right-inverse-of (Eq.extensionality-isomorphism ext₂) _ ⟩∎

             cong  h x  Embedding.to (B₁↣B₂ x) (h x)) f≡g              )

-- Π preserves all kinds of functions in its second argument (in some
-- cases assuming extensionality).

∀-cong :
   {k a b₁ b₂} 
  Extensionality? k a (b₁  b₂) 
  {A : Type a} {B₁ : A  Type b₁} {B₂ : A  Type b₂} 
  (∀ x  B₁ x ↝[ k ] B₂ x) 
  ((x : A)  B₁ x) ↝[ k ] ((x : A)  B₂ x)
∀-cong {implication}         = λ _  ∀-cong-→
∀-cong {logical-equivalence} = λ _  L.∀-cong
∀-cong {injection}           = ∀-cong-inj
∀-cong {embedding}           = ∀-cong-emb
∀-cong {surjection}          = λ ext  ∀-cong-↠ ext
∀-cong {bijection}           = ∀-cong-bij
∀-cong {equivalence}         = ∀-cong-eq
∀-cong {equivalenceᴱ}        = ∀-cong-eqᴱ

-- The implicit variant of Π preserves all kinds of functions in its
-- second argument (in some cases assuming extensionality).

implicit-∀-cong :
   {k a b₁ b₂} 
  Extensionality? k a (b₁  b₂) 
  {A : Type a} {B₁ : A  Type b₁} {B₂ : A  Type b₂} 
  (∀ {x}  B₁ x ↝[ k ] B₂ x) 
  ({x : A}  B₁ x) ↝[ k ] ({x : A}  B₂ x)
implicit-∀-cong ext {A} {B₁} {B₂} B₁↝B₂ =
  ({x : A}  B₁ x)  ↔⟨ Bijection.implicit-Π↔Π 
  ((x : A)  B₁ x)  ↝⟨ ∀-cong ext  _  B₁↝B₂) 
  ((x : A)  B₂ x)  ↔⟨ inverse Bijection.implicit-Π↔Π ⟩□
  ({x : A}  B₂ x)  

-- Two generalisations of ∀-cong for non-dependent functions.

Π-cong-contra-→ :
   {a₁ a₂ b₁ b₂}
    {A₁ : Type a₁} {A₂ : Type a₂}
    {B₁ : A₁  Type b₁} {B₂ : A₂  Type b₂} 
  (A₂→A₁ : A₂  A₁) 
  (∀ x  B₁ (A₂→A₁ x)  B₂ x) 
  ((x : A₁)  B₁ x)  ((x : A₂)  B₂ x)
Π-cong-contra-→ {B₁ = B₁} {B₂} A₂→A₁ B₁→B₂ f x =
                $⟨ f (A₂→A₁ x) 
  B₁ (A₂→A₁ x)  ↝⟨ B₁→B₂ x 
  B₂ x          

Π-cong-→ :
   {a₁ a₂ b₁ b₂}
    {A₁ : Type a₁} {A₂ : Type a₂}
    {B₁ : A₁  Type b₁} {B₂ : A₂  Type b₂} 
  (A₁↠A₂ : A₁  A₂) 
  (∀ x  B₁ x  B₂ (_↠_.to A₁↠A₂ x)) 
  ((x : A₁)  B₁ x)  ((x : A₂)  B₂ x)
Π-cong-→ {B₁ = B₁} {B₂} A₁↠A₂ B₁→B₂ f x =
                                        $⟨ f (_↠_.from A₁↠A₂ x)