------------------------------------------------------------------------
-- The Agda standard library
--
-- Equivalence (coinhabitance)
------------------------------------------------------------------------

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

module Function.Equivalence where

open import Function using (flip)
open import Function.Equality as F
  using (_⟶_; _⟨$⟩_) renaming (_∘_ to _⟪∘⟫_)
open import Level
open import Relation.Binary
import Relation.Binary.PropositionalEquality as P

------------------------------------------------------------------------
-- Setoid equivalence

record Equivalence {f₁ f₂ t₁ t₂}
                   (From : Setoid f₁ f₂) (To : Setoid t₁ t₂) :
                   Set (f₁  f₂  t₁  t₂) where
  field
    to   : From  To
    from : To  From

------------------------------------------------------------------------
-- The set of all equivalences between two sets (i.e. equivalences
-- with propositional equality)

infix 3 _⇔_

_⇔_ :  {f t}  Set f  Set t  Set _
From  To = Equivalence (P.setoid From) (P.setoid To)

equivalence :  {f t} {From : Set f} {To : Set t} 
              (From  To)  (To  From)  From  To
equivalence to from = record
  { to   = P.→-to-⟶ to
  ; from = P.→-to-⟶ from
  }

------------------------------------------------------------------------
-- Equivalence is an equivalence relation

-- Identity and composition (reflexivity and transitivity).

id :  {s₁ s₂}  Reflexive (Equivalence {s₁} {s₂})
id {x = S} = record
  { to   = F.id
  ; from = F.id
  }

infixr 9 _∘_

_∘_ :  {f₁ f₂ m₁ m₂ t₁ t₂} 
      TransFlip (Equivalence {f₁} {f₂} {m₁} {m₂})
                (Equivalence {m₁} {m₂} {t₁} {t₂})
                (Equivalence {f₁} {f₂} {t₁} {t₂})
f  g = record
  { to   = to   f ⟪∘⟫ to   g
  ; from = from g ⟪∘⟫ from f
  } where open Equivalence

-- Symmetry.

sym :  {f₁ f₂ t₁ t₂} 
      Sym (Equivalence {f₁} {f₂} {t₁} {t₂})
          (Equivalence {t₁} {t₂} {f₁} {f₂})
sym eq = record
  { from       = to
  ; to         = from
  } where open Equivalence eq

-- For fixed universe levels we can construct setoids.

setoid : (s₁ s₂ : Level)  Setoid (suc (s₁  s₂)) (s₁  s₂)
setoid s₁ s₂ = record
  { Carrier       = Setoid s₁ s₂
  ; _≈_           = Equivalence
  ; isEquivalence = record
    { refl  = id
    ; sym   = sym
    ; trans = flip _∘_
    }
  }

⇔-setoid : ( : Level)  Setoid (suc ) 
⇔-setoid  = record
  { Carrier       = Set 
  ; _≈_           = _⇔_
  ; isEquivalence = record
    { refl  = id
    ; sym   = sym
    ; trans = flip _∘_
    }
  }

------------------------------------------------------------------------
-- Transformations

map :  {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂}
        {f₁′ f₂′ t₁′ t₂′}
        {From′ : Setoid f₁′ f₂′} {To′ : Setoid t₁′ t₂′} 
      ((From  To)  (From′  To′)) 
      ((To  From)  (To′  From′)) 
      Equivalence From To  Equivalence From′ To′
map t f eq = record { to = t to; from = f from }
  where open Equivalence eq

zip :  {f₁₁ f₂₁ t₁₁ t₂₁}
        {From₁ : Setoid f₁₁ f₂₁} {To₁ : Setoid t₁₁ t₂₁}
        {f₁₂ f₂₂ t₁₂ t₂₂}
        {From₂ : Setoid f₁₂ f₂₂} {To₂ : Setoid t₁₂ t₂₂}
        {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} 
      ((From₁  To₁)  (From₂  To₂)  (From  To)) 
      ((To₁  From₁)  (To₂  From₂)  (To  From)) 
      Equivalence From₁ To₁  Equivalence From₂ To₂ 
      Equivalence From To
zip t f eq₁ eq₂ =
  record { to = t (to eq₁) (to eq₂); from = f (from eq₁) (from eq₂) }
  where open Equivalence