------------------------------------------------------------------------
-- Bijections
------------------------------------------------------------------------

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

open import Equality

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

open Derived-definitions-and-properties eq
open import Equality.Decision-procedures eq
open import H-level eq
open import Injection eq using (Injective; _↣_)
open import Logical-equivalence using (_⇔_)
open import Prelude as P hiding (id) renaming (_∘_ to _⊚_)
open import Surjection eq as Surjection using (_↠_; module _↠_)

------------------------------------------------------------------------
-- Bijections

infix 0 _↔_

record _↔_ {f t} (From : Set f) (To : Set t) : Set (f  t) where
  field
    surjection : From  To

  open _↠_ surjection

  field
    left-inverse-of :  x  from (to x)  x

  injective : Injective to
  injective {x = x} {y = y} to-x≡to-y =
    x            ≡⟨ sym (left-inverse-of x) 
    from (to x)  ≡⟨ cong from to-x≡to-y 
    from (to y)  ≡⟨ left-inverse-of y ⟩∎
    y            

  injection : From  To
  injection = record
    { to        = to
    ; injective = injective
    }

  -- A lemma.

  to-from :  {x y}  to x  y  from y  x
  to-from {x} {y} to-x≡y =
    from y       ≡⟨ cong from $ sym to-x≡y 
    from (to x)  ≡⟨ left-inverse-of x ⟩∎
    x            

  open _↠_ surjection public

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

-- _↔_ is an equivalence relation.

id :  {a} {A : Set a}  A  A
id = record
  { surjection      = Surjection.id
  ; left-inverse-of = refl
  }

inverse :  {a b} {A : Set a} {B : Set b}  A  B  B  A
inverse A↔B = record
  { surjection = record
    { logical-equivalence = Logical-equivalence.inverse
                              logical-equivalence
    ; right-inverse-of    = left-inverse-of
    }
  ; left-inverse-of = right-inverse-of
  } where open _↔_ A↔B

infixr 9 _∘_

_∘_ :  {a b c} {A : Set a} {B : Set b} {C : Set c} 
      B  C  A  B  A  C
f  g = record
  { surjection      = Surjection._∘_ (surjection f) (surjection g)
  ; left-inverse-of = from∘to
  }
  where
  open _↔_

  abstract
    from∘to :  x  from g (from f (to f (to g x)))  x
    from∘to = λ x 
      from g (from f (to f (to g x)))  ≡⟨ cong (from g) (left-inverse-of f (to g x)) 
      from g (to g x)                  ≡⟨ left-inverse-of g x ⟩∎
      x                                

-- "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 □

------------------------------------------------------------------------
-- More lemmas

-- Equality between pairs can be expressed as a pair of equalities.

Σ-≡,≡↔≡ :  {a b} {A : Set a} {B : A  Set b} {p₁ p₂ : Σ A B} 
          ( λ (p : proj₁ p₁  proj₁ p₂) 
             subst B p (proj₂ p₁)  proj₂ p₂) 
          (p₁  p₂)
Σ-≡,≡↔≡ {A = A} {B} {p₁} {p₂} = record
  { surjection = record
    { logical-equivalence = record
      { to   = to
      ; from = from
      }
    ; right-inverse-of = to∘from
    }
  ; left-inverse-of = from∘to
  }
  where
  from-P = λ {p₁ p₂ : Σ A B} (_ : p₁  p₂) 
              λ (p : proj₁ p₁  proj₁ p₂) 
               subst B p (proj₂ p₁)  proj₂ p₂

  from : {p₁ p₂ : Σ A B} 
         p₁  p₂ 
          λ (p : proj₁ p₁  proj₁ p₂) 
           subst B p (proj₂ p₁)  proj₂ p₂
  from = Σ-≡,≡←≡

  to : {p₁ p₂ : Σ A B} 
       ( λ (p : proj₁ p₁  proj₁ p₂) 
          subst B p (proj₂ p₁)  proj₂ p₂) 
       p₁  p₂
  to = uncurry Σ-≡,≡→≡

  abstract

    to∘from :  eq  to (from {p₁ = p₁} {p₂ = p₂} eq)  eq
    to∘from = elim  p≡q  to (from p≡q)  p≡q) λ x 
      let lem = subst-refl B (proj₂ x) in
      to (from (refl x))                          ≡⟨ cong to (elim-refl from-P _) 
      to (refl (proj₁ x) , lem)                   ≡⟨ Σ-≡,≡→≡-reflˡ _ 
      cong (_,_ (proj₁ x)) (trans (sym lem) lem)  ≡⟨ cong (cong (_,_ (proj₁ x))) $ trans-symˡ lem 
      cong (_,_ (proj₁ x)) (refl (proj₂ x))       ≡⟨ cong-refl (_,_ (proj₁ x)) {x = proj₂ x} ⟩∎
      refl x                                      

    from∘to :  p  from (to {p₁ = p₁} {p₂ = p₂} p)  p
    from∘to p = elim
       {x₁ x₂} x₁≡x₂ 
          {y₁ y₂} (y₁′≡y₂ : subst B x₁≡x₂ y₁  y₂) 
         from (to (x₁≡x₂ , y₁′≡y₂))  (x₁≡x₂ , y₁′≡y₂))
       x {y₁} y₁′≡y₂  elim
          {y₁ y₂} (y₁≡y₂ : y₁  y₂) 
            (y₁′≡y₂ : subst B (refl x) y₁  y₂) 
            y₁≡y₂  trans (sym $ subst-refl B y₁) y₁′≡y₂ 
            from (to (refl x , y₁′≡y₂))  (refl x , y₁′≡y₂))
          y y′≡y eq 
          let lem = subst-refl B y in
          from (to (refl x , y′≡y))                   ≡⟨ cong from $ Σ-≡,≡→≡-reflˡ _ 
          from (cong (_,_ x) (trans (sym lem) y′≡y))  ≡⟨ cong (from  cong (_,_ x)) $ sym eq 
          from (cong (_,_ x) (refl y))                ≡⟨ cong from $ cong-refl (_,_ x) {x = y} 
          from (refl (x , y))                         ≡⟨ elim-refl from-P _ 
          (refl x , lem)                              ≡⟨ cong (_,_ (refl x)) (
             lem                                           ≡⟨ sym $ trans-reflʳ _ 
             trans lem (refl _)                            ≡⟨ cong (trans lem) eq 
             trans lem (trans (sym lem) y′≡y)              ≡⟨ sym $ trans-assoc _ _ _ 
             trans (trans lem (sym lem)) y′≡y              ≡⟨ cong  p  trans p y′≡y) $ trans-symʳ lem 
             trans (refl _) y′≡y                           ≡⟨ trans-reflˡ _ ⟩∎
             y′≡y                                          ) ⟩∎
          (refl x , y′≡y)                             )
         (trans (sym $ subst-refl B y₁) y₁′≡y₂)
         y₁′≡y₂
         (refl _))
      (proj₁ p) (proj₂ p)

-- Equalities are closed, in a strong sense, under applications of
-- certain injections (at least inj₁ and inj₂).

≡↔inj₁≡inj₁ :  {a b} {A : Set a} {B : Set b} {x y : A} 
              (x  y)  _≡_ {A = A  B} (inj₁ x) (inj₁ y)
≡↔inj₁≡inj₁ {A = A} {B} {x} {y} = record
  { surjection = record
    { logical-equivalence = record
      { to   = to
      ; from = from
      }
    ; right-inverse-of = to∘from
    }
  ; left-inverse-of = from∘to
  }
  where
  to : x  y  _≡_ {A = A  B} (inj₁ x) (inj₁ y)
  to = cong inj₁

  from = ⊎.cancel-inj₁

  abstract

    to∘from :  ix≡iy  to (from ix≡iy)  ix≡iy
    to∘from ix≡iy =
      cong inj₁ (⊎.cancel-inj₁ ix≡iy)              ≡⟨ cong-∘ inj₁ [ P.id , const x ] ix≡iy 
      cong f ix≡iy                                 ≡⟨ cong-roughly-id f p ix≡iy _ _ f≡id 
      trans (refl _) (trans ix≡iy $ sym (refl _))  ≡⟨ trans-reflˡ _ 
      trans ix≡iy (sym $ refl _)                   ≡⟨ cong (trans ix≡iy) sym-refl 
      trans ix≡iy (refl _)                         ≡⟨ trans-reflʳ _ ⟩∎
      ix≡iy                                        
      where
      f : A  B  A  B
      f = inj₁  [ P.id , const x ]

      p : A  B  Bool
      p = [ const true , const false ]

      f≡id :  z  T (p z)  f z  z
      f≡id (inj₁ x) = const (refl (inj₁ x))
      f≡id (inj₂ y) = ⊥-elim

    from∘to :  x≡y  from (to x≡y)  x≡y
    from∘to x≡y =
      cong [ P.id , const x ] (cong inj₁ x≡y)  ≡⟨ cong-∘ [ P.id , const x ] inj₁ _ 
      cong P.id x≡y                            ≡⟨ sym $ cong-id _ ⟩∎
      x≡y                                      

≡↔inj₂≡inj₂ :  {a b} {A : Set a} {B : Set b} {x y : B} 
              (x  y)  _≡_ {A = A  B} (inj₂ x) (inj₂ y)
≡↔inj₂≡inj₂ {A = A} {B} {x} {y} = record
  { surjection = record
    { logical-equivalence = record
      { to   = to
      ; from = from
      }
    ; right-inverse-of = to∘from
    }
  ; left-inverse-of = from∘to
  }
  where
  to : x  y  _≡_ {A = A  B} (inj₂ x) (inj₂ y)
  to = cong inj₂

  from = ⊎.cancel-inj₂

  abstract

    to∘from :  ix≡iy  to (from ix≡iy)  ix≡iy
    to∘from ix≡iy =
      cong inj₂ (⊎.cancel-inj₂ ix≡iy)              ≡⟨ cong-∘ inj₂ [ const x , P.id ] ix≡iy 
      cong f ix≡iy                                 ≡⟨ cong-roughly-id f p ix≡iy _ _ f≡id 
      trans (refl _) (trans ix≡iy $ sym (refl _))  ≡⟨ trans-reflˡ _ 
      trans ix≡iy (sym $ refl _)                   ≡⟨ cong (trans ix≡iy) sym-refl 
      trans ix≡iy (refl _)                         ≡⟨ trans-reflʳ _ ⟩∎
      ix≡iy                                        
      where
      f : A  B  A  B
      f = inj₂  [ const x , P.id ]

      p : A  B  Bool
      p = [ const false , const true ]

      f≡id :  z  T (p z)  f z  z
      f≡id (inj₁ x) = ⊥-elim
      f≡id (inj₂ y) = const (refl (inj₂ y))

    from∘to :  x≡y  from (to x≡y)  x≡y
    from∘to x≡y =
      cong [ const x , P.id ] (cong inj₂ x≡y)  ≡⟨ cong-∘ [ const x , P.id ] inj₂ _ 
      cong P.id x≡y                            ≡⟨ sym $ cong-id _ ⟩∎
      x≡y                                      

-- Decidable equality respects bijections.

decidable-equality-respects :
   {a b} {A : Set a} {B : Set b} 
  A  B  Decidable-equality A  Decidable-equality B
decidable-equality-respects A↔B _≟A_ x y =
  ⊎-map (_↔_.injective (inverse A↔B))
                        from-x≢from-y  from-x≢from-y  cong from)
        (from x ≟A from y)
  where open _↔_ A↔B

-- All contractible types are isomorphic.

contractible-isomorphic :
   {a b} {A : Set a} {B : Set b} 
  Contractible A  Contractible B  A  B
contractible-isomorphic {A} {B} cA cB = record
  { surjection = record
    { logical-equivalence = record
      { to   = const (proj₁ cB)
      ; from = const (proj₁ cA)
      }
    ; right-inverse-of = proj₂ cB
    }
  ; left-inverse-of = proj₂ cA
  }

-- Implicit and explicit Πs are isomorphic.

implicit-Π↔Π :  {a b} {A : Set a} {B : A  Set b} 
               ({x : A}  B x)  ((x : A)  B x)
implicit-Π↔Π = record
  { surjection = record
    { logical-equivalence = record
      { to   = λ f x  f {x}
      ; from = λ f {x}  f x
      }
    ; right-inverse-of = refl
    }
  ; left-inverse-of = refl
  }

-- A lifted set is isomorphic to the underlying one.

↑↔ :  {a b} {A : Set a}   b A  A
↑↔ {b = b} {A} = record
  { surjection = record
    { logical-equivalence = record
      { to   = lower
      ; from = lift
      }
    ; right-inverse-of = refl
    }
  ; left-inverse-of = refl
  }