------------------------------------------------------------------------
-- The Agda standard library
--
-- Inverses
------------------------------------------------------------------------

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

module Function.Inverse where

open import Level
open import Function using (flip)
open import Function.Bijection hiding (id; _∘_; bijection)
open import Function.Equality as F
  using (_⟶_) renaming (_∘_ to _⟪∘⟫_)
open import Function.LeftInverse as Left hiding (id; _∘_)
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as P using (_≗_; _≡_)
open import Relation.Unary using (Pred)

------------------------------------------------------------------------
-- Inverses

record _InverseOf_ {f₁ f₂ t₁ t₂}
                   {From : Setoid f₁ f₂} {To : Setoid t₁ t₂}
                   (from : To  From) (to : From  To) :
                   Set (f₁  f₂  t₁  t₂) where
  field
    left-inverse-of  : from LeftInverseOf  to
    right-inverse-of : from RightInverseOf to

------------------------------------------------------------------------
-- The set of all inverses between two setoids

record Inverse {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
    inverse-of : from InverseOf to

  open _InverseOf_ inverse-of public

  left-inverse : LeftInverse From To
  left-inverse = record
    { to              = to
    ; from            = from
    ; left-inverse-of = left-inverse-of
    }

  open LeftInverse left-inverse public
    using (injective; injection)

  bijection : Bijection From To
  bijection = record
    { to        = to
    ; bijective = record
      { injective  = injective
      ; surjective = record
        { from             = from
        ; right-inverse-of = right-inverse-of
        }
      }
    }

  open Bijection bijection public
    using (equivalence; surjective; surjection; right-inverse;
           to-from; from-to)

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

infix 3 _↔_ _↔̇_

_↔_ :  {f t}  Set f  Set t  Set _
From  To = Inverse (P.setoid From) (P.setoid To)

_↔̇_ :  {i f t} {I : Set i}  Pred I f  Pred I t  Set _
From ↔̇ To =  {i}  From i  To i

inverse :  {f t} {From : Set f} {To : Set t} 
          (to : From  To) (from : To  From) 
          (∀ x  from (to x)  x) 
          (∀ x  to (from x)  x) 
          From  To
inverse to from from∘to to∘from = record
  { to   = P.→-to-⟶ to
  ; from = P.→-to-⟶ from
  ; inverse-of = record
    { left-inverse-of  = from∘to
    ; right-inverse-of = to∘from
    }
  }

------------------------------------------------------------------------
-- If two setoids are in bijective correspondence, then there is an
-- inverse between them

fromBijection :
   {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} 
  Bijection From To  Inverse From To
fromBijection b = record
  { to         = Bijection.to b
  ; from       = Bijection.from b
  ; inverse-of = record
    { left-inverse-of  = Bijection.left-inverse-of b
    ; right-inverse-of = Bijection.right-inverse-of b
    }
  }

------------------------------------------------------------------------
-- Inverse is an equivalence relation

-- Reflexivity

id :  {s₁ s₂}  Reflexive (Inverse {s₁} {s₂})
id {x = S} = record
  { to         = F.id
  ; from       = F.id
  ; inverse-of = record
    { left-inverse-of  = LeftInverse.left-inverse-of id′
    ; right-inverse-of = LeftInverse.left-inverse-of id′
    }
  } where id′ = Left.id {S = S}

-- Transitivity

infixr 9 _∘_

_∘_ :  {f₁ f₂ m₁ m₂ t₁ t₂} 
      TransFlip (Inverse {f₁} {f₂} {m₁} {m₂})
                (Inverse {m₁} {m₂} {t₁} {t₂})
                (Inverse {f₁} {f₂} {t₁} {t₂})
f  g = record
  { to         = to   f ⟪∘⟫ to   g
  ; from       = from g ⟪∘⟫ from f
  ; inverse-of = record
    { left-inverse-of  = LeftInverse.left-inverse-of (Left._∘_ (left-inverse  f) (left-inverse  g))
    ; right-inverse-of = LeftInverse.left-inverse-of (Left._∘_ (right-inverse g) (right-inverse f))
    }
  } where open Inverse

-- Symmetry.

sym :  {f₁ f₂ t₁ t₂} 
      Sym (Inverse {f₁} {f₂} {t₁} {t₂}) (Inverse {t₁} {t₂} {f₁} {f₂})
sym inv = record
  { from       = to
  ; to         = from
  ; inverse-of = record
    { left-inverse-of  = right-inverse-of
    ; right-inverse-of = left-inverse-of
    }
  } where open Inverse inv

------------------------------------------------------------------------
-- 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′} 
      (t : (From  To)  (From′  To′)) 
      (f : (To  From)  (To′  From′)) 
      (∀ {to from}  from InverseOf to  f from InverseOf t to) 
      Inverse From To  Inverse From′ To′
map t f pres eq = record
  { to         = t to
  ; from       = f from
  ; inverse-of = pres inverse-of
  } where open Inverse 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} 
      (t : (From₁  To₁)  (From₂  To₂)  (From  To)) 
      (f : (To₁  From₁)  (To₂  From₂)  (To  From)) 
      (∀ {to₁ from₁ to₂ from₂} 
         from₁ InverseOf to₁  from₂ InverseOf to₂ 
         f from₁ from₂ InverseOf t to₁ to₂) 
      Inverse From₁ To₁  Inverse From₂ To₂  Inverse From To
zip t f pres eq₁ eq₂ = record
  { to         = t (to   eq₁) (to   eq₂)
  ; from       = f (from eq₁) (from eq₂)
  ; inverse-of = pres (inverse-of eq₁) (inverse-of eq₂)
  } where open Inverse