{-
Theory about isomorphisms
- Definitions of [section] and [retract]
- Definition of isomorphisms ([Iso])
- Any isomorphism is an equivalence ([isoToEquiv])
-}
{-# OPTIONS --safe #-}
module Cubical.Foundations.Isomorphism where
open import Cubical.Core.Everything
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Function
open import Cubical.Foundations.Equiv.Base
private
  variable
     ℓ' : Level
    A B C : Type 
-- Section and retract
module _ { ℓ'} {A : Type } {B : Type ℓ'} where
  section : (f : A  B)  (g : B  A)  Type ℓ'
  section f g =  b  f (g b)  b
  -- NB: `g` is the retraction!
  retract : (f : A  B)  (g : B  A)  Type 
  retract f g =  a  g (f a)  a
record Iso { ℓ'} (A : Type ) (B : Type ℓ') : Type (ℓ-max  ℓ') where
  no-eta-equality
  constructor iso
  field
    fun : A  B
    inv : B  A
    rightInv : section fun inv
    leftInv  : retract fun inv
isIso : (A  B)  Type _
isIso {A = A} {B = B} f = Σ[ g  (B  A) ] Σ[ _  section f g ] retract f g
isoFunInjective : (f : Iso A B)  (x y : A)  Iso.fun f x  Iso.fun f y  x  y
isoFunInjective f x y h = sym (Iso.leftInv f x) ∙∙ cong (Iso.inv f) h ∙∙ Iso.leftInv f y
isoInvInjective : (f : Iso A B)  (x y : B)  Iso.inv f x  Iso.inv f y  x  y
isoInvInjective f x y h = sym (Iso.rightInv f x) ∙∙ cong (Iso.fun f) h ∙∙ Iso.rightInv f y
-- Any iso is an equivalence
module _ (i : Iso A B) where
  open Iso i renaming ( fun to f
                      ; inv to g
                      ; rightInv to s
                      ; leftInv to t)
  private
    module _ (y : B) (x0 x1 : A) (p0 : f x0  y) (p1 : f x1  y) where
      fill0 : I  I  A
      fill0 i = hfill  k  λ { (i = i1)  t x0 k
                               ; (i = i0)  g y })
                      (inS (g (p0 (~ i))))
      fill1 : I  I  A
      fill1 i = hfill  k  λ { (i = i1)  t x1 k
                               ; (i = i0)  g y })
                      (inS (g (p1 (~ i))))
      fill2 : I  I  A
      fill2 i = hfill  k  λ { (i = i1)  fill1 k i1
                               ; (i = i0)  fill0 k i1 })
                      (inS (g y))
      p : x0  x1
      p i = fill2 i i1
      sq : I  I  A
      sq i j = hcomp  k  λ { (i = i1)  fill1 j (~ k)
                              ; (i = i0)  fill0 j (~ k)
                              ; (j = i1)  t (fill2 i i1) (~ k)
                              ; (j = i0)  g y })
                     (fill2 i j)
      sq1 : I  I  B
      sq1 i j = hcomp  k  λ { (i = i1)  s (p1 (~ j)) k
                               ; (i = i0)  s (p0 (~ j)) k
                               ; (j = i1)  s (f (p i)) k
                               ; (j = i0)  s y k })
                      (f (sq i j))
      lemIso : (x0 , p0)  (x1 , p1)
      lemIso i .fst = p i
      lemIso i .snd = λ j  sq1 i (~ j)
  isoToIsEquiv : isEquiv f
  isoToIsEquiv .equiv-proof y .fst .fst = g y
  isoToIsEquiv .equiv-proof y .fst .snd = s y
  isoToIsEquiv .equiv-proof y .snd z = lemIso y (g y) (fst z) (s y) (snd z)

isoToEquiv : Iso A B  A  B
isoToEquiv i .fst = i .Iso.fun
isoToEquiv i .snd = isoToIsEquiv i
isoToPath : Iso A B  A  B
isoToPath {A = A} {B = B} f i =
  Glue B  { (i = i0)  (A , isoToEquiv f)
            ; (i = i1)  (B , idEquiv B) })
open Iso
invIso : Iso A B  Iso B A
fun (invIso f) = inv f
inv (invIso f) = fun f
rightInv (invIso f) = leftInv f
leftInv (invIso f)  = rightInv f
compIso : Iso A B  Iso B C  Iso A C
fun (compIso i j)       = fun j  fun i
inv (compIso i j) = inv i  inv j
rightInv (compIso i j) b = cong (fun j) (rightInv i (inv j b))  rightInv j b
leftInv (compIso i j) a = cong (inv i) (leftInv j (fun i a))  leftInv i a
composesToId→Iso : (G : Iso A B) (g : B  A)  G .fun  g  idfun B  Iso B A
fun (composesToId→Iso _ g _)             = g
inv (composesToId→Iso j _ _) = fun j
rightInv (composesToId→Iso i g path) b =
  sym (leftInv i (g (fun i b))) ∙∙ cong  g  inv i (g (fun i b))) path ∙∙ leftInv i b
leftInv (composesToId→Iso _ _ path) b i = path i b
idIso : Iso A A
fun idIso = idfun _
inv idIso = idfun _
rightInv idIso _ = refl
leftInv idIso _  = refl
compIsoIdL : (isom : Iso A B)  compIso idIso isom  isom
fun (compIsoIdL isom i) = fun isom
inv (compIsoIdL isom i) = inv isom
rightInv (compIsoIdL isom i) b = lUnit (isom .rightInv b) (~ i)
leftInv (compIsoIdL isom i) a = rUnit (isom .leftInv a) (~ i)
compIsoIdR : (isom : Iso A B)  compIso isom idIso  isom
fun (compIsoIdR isom i) = fun isom
inv (compIsoIdR isom i) = inv isom
rightInv (compIsoIdR isom i) b = rUnit (isom .rightInv b) (~ i)
leftInv (compIsoIdR isom i) a = lUnit (isom .leftInv a) (~ i)
LiftIso : Iso A (Lift {i = } {j = ℓ'} A)
fun LiftIso = lift
inv LiftIso = lower
rightInv LiftIso _ = refl
leftInv LiftIso _  = refl
isContr→Iso : isContr A  isContr B  Iso A B
fun (isContr→Iso _ Bctr) _ = Bctr .fst
inv (isContr→Iso Actr _) _ = Actr .fst
rightInv (isContr→Iso _ Bctr) = Bctr .snd
leftInv (isContr→Iso Actr _)  = Actr .snd
isContr→Iso' : isContr A  isContr B  (A  B)  Iso A B
fun (isContr→Iso' _ Bctr f) = f
inv (isContr→Iso' Actr _ _) _ = Actr .fst
rightInv (isContr→Iso' _ Bctr f) = isContr→isProp Bctr _
leftInv (isContr→Iso' Actr _ _)  = Actr .snd
isProp→Iso :  (Aprop : isProp A) (Bprop : isProp B) (f : A  B) (g : B  A)  Iso A B
fun (isProp→Iso _ _ f _) = f
inv (isProp→Iso _ _ _ g) = g
rightInv (isProp→Iso _ Bprop f g) b = Bprop (f (g b)) b
leftInv (isProp→Iso Aprop _ f g) a  = Aprop (g (f a)) a
domIso :  {} {C : Type }  Iso A B  Iso (A  C) (B  C)
fun (domIso e) f b = f (inv e b)
inv (domIso e) f a = f (fun e a)
rightInv (domIso e) f i x = f (rightInv e x i)
leftInv (domIso e) f i x = f (leftInv e x i)
-- Helpful notation
_Iso⟨_⟩_ :  { ℓ' ℓ''} {B : Type ℓ'} {C : Type ℓ''} (X : Type )  Iso X B  Iso B C  Iso X C
_ Iso⟨ f  g = compIso f g
_∎Iso :  {} (A : Type )  Iso A A
A ∎Iso = idIso {A = A}
infixr  0 _Iso⟨_⟩_
infix   1 _∎Iso
codomainIsoDep :  { ℓ' ℓ''} {A : Type } {B : A  Type ℓ'} {C : A  Type ℓ''}
                  ((a : A)  Iso (B a) (C a))
                  Iso ((a : A)  B a) ((a : A)  C a)
fun (codomainIsoDep is) f a = fun (is a) (f a)
inv (codomainIsoDep is) f a = inv (is a) (f a)
rightInv (codomainIsoDep is) f = funExt λ a  rightInv (is a) (f a)
leftInv (codomainIsoDep is) f = funExt λ a  leftInv (is a) (f a)
codomainIso :  { ℓ' ℓ''} {A : Type } {B : Type ℓ'} {C : Type ℓ''}
            Iso B C
            Iso (A  B) (A  C)
codomainIso z = codomainIsoDep λ _  z

Iso≡Set : isSet A  isSet B  (f g : Iso A B)
         ((x : A)  f .fun x  g .fun x)
         ((x : B)  f .inv x  g .inv x)
         f  g
fun (Iso≡Set hA hB f g hfun hinv i) x = hfun x i
inv (Iso≡Set hA hB f g hfun hinv i) x = hinv x i
rightInv (Iso≡Set hA hB f g hfun hinv i) x j =
  isSet→isSet' hB (rightInv f x) (rightInv g x)  i  hfun (hinv x i) i) refl i j
leftInv (Iso≡Set hA hB f g hfun hinv i) x j =
  isSet→isSet' hA (leftInv f x) (leftInv g x)  i  hinv (hfun x i) i) refl i j