------------------------------------------------------------------------
-- Closure properties for h-levels
------------------------------------------------------------------------

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

-- Partly based on Voevodsky's work on so-called univalent
-- foundations.

open import Equality

module H-level.Closure
  {reflexive} (eq :  {a p}  Equality-with-J a p reflexive) where

open import Bijection eq as Bijection hiding (id; _∘_)
open Derived-definitions-and-properties eq
import Equality.Decidable-UIP eq as DUIP
open import Equality.Decision-procedures eq
open import H-level eq
open import Logical-equivalence hiding (id; _∘_)
open import Prelude
open import Surjection eq as Surjection hiding (id; _∘_)

------------------------------------------------------------------------
-- The unit type

-- The unit type is contractible.

⊤-contractible : Contractible 
⊤-contractible = (_ , λ _  refl _)

abstract

  -- A type is contractible iff it is in bijective correspondence with
  -- the unit type.

  contractible⇔⊤↔ :  {a} {A : Set a}  Contractible A  (  A)
  contractible⇔⊤↔ = λ {a} {A}  record
    { to   = contractible-isomorphic ⊤-contractible
    ; from = λ ⊤↔A  respects-surjection (_↔_.surjection ⊤↔A) 0
                                         ⊤-contractible
    }

------------------------------------------------------------------------
-- The empty type

abstract

  -- The empty type is not contractible.

  ¬-⊥-contractible :  {}  ¬ Contractible ( {= })
  ¬-⊥-contractible = ⊥-elim  proj₁

  -- The empty type is propositional.

  ⊥-propositional :  {}  Is-proposition ( {= })
  ⊥-propositional =
    _⇔_.from propositional⇔irrelevant  x  ⊥-elim x)

  -- Thus any uninhabited type is also propositional.

  ⊥↔uninhabited :  {a } {A : Set a}  ¬ A   {= }  A
  ⊥↔uninhabited ¬A = record
    { surjection = record
      { logical-equivalence = record
        { to   = ⊥-elim
        ; from = ⊥-elim  ¬A
        }
      ; right-inverse-of = ⊥-elim  ¬A
      }
    ; left-inverse-of = λ ()
    }

  uninhabited-propositional :  {a} {A : Set a} 
                              ¬ A  Is-proposition A
  uninhabited-propositional ¬A =
    respects-surjection (_↔_.surjection $ ⊥↔uninhabited {= # 0} ¬A) 1
                        ⊥-propositional

------------------------------------------------------------------------
-- Booleans

abstract

  -- The booleans are not propositional.

  ¬-Bool-propositional : ¬ Is-proposition Bool
  ¬-Bool-propositional propositional =
    Bool.true≢false $
    (_⇔_.to propositional⇔irrelevant propositional) true false

  -- The booleans form a set.

  Bool-set : Is-set Bool
  Bool-set = decidable⇒set Bool._≟_

------------------------------------------------------------------------
-- Natural numbers

abstract

  -- ℕ is not propositional.

  ¬-ℕ-propositional : ¬ Is-proposition 
  ¬-ℕ-propositional ℕ-prop =
    ℕ.0≢+ $ _⇔_.to propositional⇔irrelevant ℕ-prop 0 1

  -- ℕ is a set.

  ℕ-set : Is-set 
  ℕ-set = decidable⇒set ℕ._≟_

------------------------------------------------------------------------
-- Π-types

-- Closure of contractibility under Π A is logically equivalent to
-- having extensional equality for functions from A.

Π-closure-contractible⇔extensionality :
   {a b} {A : Set a} 
  ({B : A  Set b} 
   (∀ x  Contractible (B x))  Contractible ((x : A)  B x)) 
  ({B : A  Set b}  Extensionality′ A B)
Π-closure-contractible⇔extensionality {b = b} {A} = record
  { to   = 
  ; from = λ ext cB 
      ((λ x  proj₁ (cB x)) , λ f  ext λ x  proj₂ (cB x) (f x))
  }
  where
   : ({B : A  Set b} 
       (∀ x  Contractible (B x))  Contractible ((x : A)  B x)) 
      (∀ {B}  Extensionality′ A B)
   closure {B} {f} {g} f≡g =
    f                                     ≡⟨ sym (cong  c  λ x  proj₁ (c x)) $
                                               proj₂ contractible  x  (f x , f≡g x))) 
     x  proj₁ (proj₁ contractible x))  ≡⟨ cong  c  λ x  proj₁ (c x)) $
                                               proj₂ contractible  x  (g x , refl (g x))) ⟩∎
    g                                     
    where
    contractible : Contractible ((x : A)  Singleton (g x))
    contractible = closure (singleton-contractible  g)

abstract

  -- Given (generalised) extensionality one can define an
  -- extensionality proof which is well-behaved.

  extensionality⇒well-behaved-extensionality :
     {a b} {A : Set a} 
    ({B : A  Set b}  Extensionality′ A B) 
    {B : A  Set b}  Well-behaved-extensionality A B
  extensionality⇒well-behaved-extensionality {A = A} ext {B} =
     {_}  ext′) , λ f 
      ext′ (refl  f)  ≡⟨ trans-symˡ _ ⟩∎
      refl f           
    where
    ext′ : Extensionality′ A B
    ext′ = to (from ext)
      where open _⇔_ Π-closure-contractible⇔extensionality

-- A potential inverse of extensionality. (See Equivalence for a proof
-- which shows that this function has an inverse, assuming
-- extensionality.)

ext⁻¹ :  {a b} {A : Set a} {B : A  Set b} {f g : (x : A)  B x} 
        f  g  (∀ x  f x  g x)
ext⁻¹ f≡g = λ x  cong  h  h x) f≡g

abstract

  -- "Evaluation rule" for ext⁻¹.

  ext⁻¹-refl :  {a b} {A : Set a} {B : A  Set b}
               (f : (x : A)  B x) {x} 
               ext⁻¹ (refl f) x  refl (f x)
  ext⁻¹-refl f {x} = cong-refl  h  h x) {x = f}

  -- Given extensionality there is a (split) surjection from
  -- ∀ x → f x ≡ g x to f ≡ g.

  ext-surj :  {a b} {A : Set a} 
             ({B : A  Set b}  Extensionality′ A B) 
             {B : A  Set b} {f g : (x : A)  B x} 
             (∀ x  f x  g x)  (f  g)
  ext-surj {b = b} {A} ext {B} = record
    { logical-equivalence = record
      { to   = to
      ; from = ext⁻¹
      }
    ; right-inverse-of =
        elim  {f g} f≡g  to (ext⁻¹ f≡g)  f≡g) λ h 
          proj₁ ext′ (ext⁻¹ (refl h))  ≡⟨ cong (proj₁ ext′) (proj₁ ext′ λ _ 
                                            ext⁻¹-refl h) 
          proj₁ ext′ (refl  h)        ≡⟨ proj₂ ext′ h ⟩∎
          refl h                       
    }
    where
    ext′ : {B : A  Set b}  Well-behaved-extensionality A B
    ext′ = extensionality⇒well-behaved-extensionality ext

    to : {f g : (x : A)  B x}  (∀ x  f x  g x)  f  g
    to = proj₁ ext′

  -- H-level is closed under Π A, assuming extensionality for
  -- functions from A.

  Π-closure :  {a b} {A : Set a} 
              ({B : A  Set b}  Extensionality′ A B) 
               {B : A  Set b} n 
              (∀ x  H-level n (B x))  H-level n ((x : A)  B x)
  Π-closure ext zero =
    _⇔_.from Π-closure-contractible⇔extensionality ext
  Π-closure ext (suc n) = λ h f g 
    respects-surjection (ext-surj ext) n $
      Π-closure ext n  x  h x (f x) (g x))

  -- This also applies to the implicit function space.

  implicit-Π-closure :
     {a b} {A : Set a} 
    ({B : A  Set b}  Extensionality′ A B) 
     {B : A  Set b} n 
    (∀ x  H-level n (B x))  H-level n ({x : A}  B x)
  implicit-Π-closure {A = A} ext {B} n =
    respects-surjection
      (_↔_.surjection $ Bijection.inverse implicit-Π↔Π) n 
    Π-closure ext n

  -- Negated types are propositional, assuming extensionality.

  ¬-propositional :
     {a} {A : Set a} 
    ({B : A  Set}  Extensionality′ A B) 
    Is-proposition (¬ A)
  ¬-propositional ext = Π-closure ext 1  _  ⊥-propositional)

------------------------------------------------------------------------
-- Σ-types

abstract

  -- H-level is closed under Σ.

  Σ-closure :  {a b} {A : Set a} {B : A  Set b} n 
              H-level n A  (∀ x  H-level n (B x))  H-level n (Σ A B)
  Σ-closure {A = A} {B} zero (x , irrA) hB =
    ((x , proj₁ (hB x)) , λ p 
       (x       , proj₁ (hB x))          ≡⟨ elim  {x y} _  _≡_ {A = Σ A B} (x , proj₁ (hB x))
                                                                              (y , proj₁ (hB y)))
                                                  _  refl _)
                                                 (irrA (proj₁ p)) 
       (proj₁ p , proj₁ (hB (proj₁ p)))  ≡⟨ cong (_,_ (proj₁ p)) (proj₂ (hB (proj₁ p)) (proj₂ p)) ⟩∎
       p                                 )
  Σ-closure {B = B} (suc n) hA hB = λ p₁ p₂ 
    respects-surjection (_↔_.surjection Σ-≡,≡↔≡) n $
      Σ-closure n (hA (proj₁ p₁) (proj₁ p₂))
         pr₁p₁≡pr₁p₂ 
           hB (proj₁ p₂) (subst B pr₁p₁≡pr₁p₂ (proj₂ p₁)) (proj₂ p₂))

  -- H-level is closed under _×_.

  ×-closure :  {a b} {A : Set a} {B : Set b} n 
              H-level n A  H-level n B  H-level n (A × B)
  ×-closure n hA hB = Σ-closure n hA (const hB)

------------------------------------------------------------------------
-- Lifted types

abstract

  -- All H-levels are closed under lifting.

  ↑-closure :  {a b} {A : Set a} n  H-level n A  H-level n ( b A)
  ↑-closure =
    respects-surjection (_↔_.surjection (Bijection.inverse ↑↔))

------------------------------------------------------------------------
-- W-types

-- W-types are isomorphic to Σ-types containing W-types.

W-unfolding :  {a b} {A : Set a} {B : A  Set b} 
              W A B   λ (x : A)  B x  W A B
W-unfolding = record
  { surjection = record
    { logical-equivalence = record
      { to   = λ w  head w , tail w
      ; from = uncurry sup
      }
    ; right-inverse-of = refl
    }
  ; left-inverse-of = left-inverse-of
  }
  where
  left-inverse-of : (w : W _ _)  sup (head w) (tail w)  w
  left-inverse-of (sup x f) = refl (sup x f)

abstract

  -- Equality between elements of a W-type can be proved using a pair
  -- of equalities (assuming extensionality).

  W-≡,≡↠≡ :  {a b} {A : Set a} {B : A  Set b} 
            (∀ {x} {C : B x  Set (a  b)}  Extensionality′ (B x) C) 
             {x y} {f : B x  W A B} {g : B y  W A B} 
            ( λ (p : x  y)   i  f i  g (subst B p i)) 
            (sup x f  sup y g)
  W-≡,≡↠≡ {a} {A = A} {B} ext {x} {y} {f} {g} =
    ( λ (p : x  y)   i  f i  g (subst B p i))        ↠⟨ Surjection.∃-cong lemma 
    ( λ (p : x  y)  subst  x  B x  W A B) p f  g)  ↠⟨ _↔_.surjection Σ-≡,≡↔≡ 
    (_≡_ {A =  λ (x : A)  B x  W A B} (x , f) (y , g))  ↠⟨ ↠-≡ (_↔_.surjection (Bijection.inverse (W-unfolding {A = A} {B = B}))) ⟩□
    (sup x f  sup y g)                                    
    where
    lemma : (p : x  y) 
            (∀ i  f i  g (subst B p i)) 
            (subst  x  B x  W A B) p f  g)
    lemma p = elim
       {x y} p  (f : B x  W A B) (g : B y  W A B) 
                   (∀ i  f i  g (subst B p i)) 
                   (subst  x  B x  W A B) p f  g))
       x f g 
         (∀ i  f i  g (subst B (refl x) i))        ↠⟨ subst  h  (∀ i  f i  g (h i))  (∀ i  f i  g i))
                                                              (sym (lower-extensionality₂ a ext (subst-refl B)))
                                                              Surjection.id 
         (∀ i  f i  g i)                           ↠⟨ ext-surj ext 
         (f  g)                                     ↠⟨ subst  h  (f  g)  (h  g))
                                                              (sym $ subst-refl  x'  B x'  W A B) f)
                                                              Surjection.id ⟩□
         (subst  x  B x  W A B) (refl x) f  g)  )
      p f g

  -- H-level is not closed under W.

  ¬-W-closure-contractible :  {a b} 
    ¬ (∀ {A : Set a} {B : A  Set b} 
       Contractible A  (∀ x  Contractible (B x)) 
       Contractible (W A B))
  ¬-W-closure-contractible closure =
    inhabited⇒W-empty (const (lift tt)) $
    proj₁ $
    closure (↑-closure 0 ⊤-contractible)
            (const (↑-closure 0 ⊤-contractible))

  ¬-W-closure :  {a b} 
    ¬ (∀ {A : Set a} {B : A  Set b} n 
       H-level n A  (∀ x  H-level n (B x))  H-level n (W A B))
  ¬-W-closure closure = ¬-W-closure-contractible (closure 0)

  -- However, all positive h-levels are closed under W, assuming that
  -- equality is sufficiently extensional.

  W-closure :
     {a b} {A : Set a} {B : A  Set b} 
    (∀ {x} {C : B x  Set (a  b)}  Extensionality′ (B x) C) 
     n  H-level (1 + n) A  H-level (1 + n) (W A B)
  W-closure {A = A} {B} ext n h = closure
    where
    closure : (x y : W A B)  H-level n (x  y)
    closure (sup x f) (sup y g) =
      respects-surjection (W-≡,≡↠≡ ext) n $
        Σ-closure n (h x y)  _ 
          Π-closure ext n  i  closure (f _) (g _)))

------------------------------------------------------------------------
-- H-levels

abstract

  -- Contractible is /not/ a comonad in the category of types and
  -- functions, because map cannot be defined, but we can at least
  -- define the following functions.

  counit :  {a} {A : Set a}  Contractible A  A
  counit = proj₁

  cojoin :  {a} {A : Set a} 
           ({B : A  Set a}  Extensionality′ A B) 
           Contractible A  Contractible (Contractible A)
  cojoin {A = A} ext contr = contr₃
    where
    x : A
    x = proj₁ contr

    contr₁ : Contractible (∀ y  x  y)
    contr₁ = Π-closure ext 0 (mono₁ 0 contr x)

    contr₂ : (x : A)  Contractible (∀ y  x  y)
    contr₂ x =
      subst  x  Contractible (∀ y  x  y)) (proj₂ contr x) contr₁

    contr₃ : Contractible ( λ (x : A)   y  x  y)
    contr₃ = Σ-closure 0 contr contr₂

  -- Contractible is not necessarily contractible.

  ¬-Contractible-contractible :
    ¬ ({A : Set}  Contractible (Contractible A))
  ¬-Contractible-contractible contr = proj₁ $ proj₁ $ contr {A = }

  -- Contractible is propositional (assuming extensionality).

  Contractible-propositional :
     {a} {A : Set a} 
    ({B : A  Set a}  Extensionality′ A B) 
    Is-proposition (Contractible A)
  Contractible-propositional ext =
    [inhabited⇒contractible]⇒propositional (cojoin ext)

  -- All h-levels are propositional (assuming extensionality).

  H-level-propositional :
     {a}  Extensionality a a 
     {A : Set a} n  Is-proposition (H-level n A)
  H-level-propositional     ext zero    = Contractible-propositional ext
  H-level-propositional {A} ext (suc n) =
    Π-closure ext 1 λ x 
    Π-closure ext 1 λ y 
    H-level-propositional ext {A = x  y} n

------------------------------------------------------------------------
-- Binary sums

abstract

  -- Binary sums can be expressed using Σ and Bool (with large
  -- elimination).

  sum-as-pair :  {a b} {A : Set a} {B : Set b} 
                (A  B)  ( λ x  if x then  b A else  a B)
  sum-as-pair {a} {b} {A} {B} = record
    { surjection = record
      { logical-equivalence = record
        { to   = to
        ; from = from
        }
      ; right-inverse-of = to∘from
      }
    ; left-inverse-of = [ refl  inj₁ {B = B} , refl  inj₂ {A = A} ]
    }
    where
    to : A  B  ( λ x  if x then  b A else  a B)
    to = [ _,_ true  lift , _,_ false  lift ]

    from : ( λ x  if x then  b A else  a B)  A  B
    from (true  , x) = inj₁ $ lower x
    from (false , y) = inj₂ $ lower y

    to∘from : (y :  λ x  if x then  b A else  a B) 
              to (from y)  y
    to∘from (true  , x) = refl _
    to∘from (false , y) = refl _

  -- H-level is not closed under _⊎_.

  ¬-⊎-propositional :  {a b} {A : Set a} {B : Set b} 
                      A  B  ¬ Is-proposition (A  B)
  ¬-⊎-propositional {A = A} {B} x y hA⊎B =
    ⊎.inj₁≢inj₂ {A = A} {B = B} $ proj₁ $ hA⊎B (inj₁ x) (inj₂ y)

  ¬-⊎-closure :  {a b} 
    ¬ (∀ {A : Set a} {B : Set b} n 
       H-level n A  H-level n B  H-level n (A  B))
  ¬-⊎-closure ⊎-closure =
    ¬-⊎-propositional (lift tt) (lift tt) $
    mono₁ 0 $
    ⊎-closure 0 (↑-closure 0 ⊤-contractible)
                (↑-closure 0 ⊤-contractible)

  -- However, all levels greater than or equal to 2 are closed under
  -- _⊎_.

  ⊎-closure :
     {a b} {A : Set a} {B : Set b} n 
    H-level (2 + n) A  H-level (2 + n) B  H-level (2 + n) (A  B)
  ⊎-closure {a} {b} {A} {B} n hA hB =
    respects-surjection
      (_↔_.surjection $ Bijection.inverse sum-as-pair)
      (2 + n)
      (Σ-closure (2 + n) Bool-2+n if-2+n)
    where
    Bool-2+n : H-level (2 + n) Bool
    Bool-2+n = mono (m≤m+n 2 n) Bool-set

    if-2+n :  x  H-level (2 + n) (if x then  b A else  a B)
    if-2+n true  = respects-surjection
                     (_↔_.surjection $ Bijection.inverse ↑↔)
                     (2 + n) hA
    if-2+n false = respects-surjection
                     (_↔_.surjection $ Bijection.inverse ↑↔)
                     (2 + n) hB

  -- Furthermore Is-proposition is closed under Dec (assuming
  -- extensionality).

  Dec-closure-propositional :
     {a} {A : Set a} 
    ({B : A  Set}  Extensionality′ A B) 
    Is-proposition A  Is-proposition (Dec A)
  Dec-closure-propositional {A = A} ext p =
    _⇔_.from propositional⇔irrelevant irrelevant
    where
    irrelevant : Proof-irrelevant (Dec A)
    irrelevant (inj₁  a) (inj₁  a′) = cong (inj₁ {B = ¬ A}) $ proj₁ $ p a a′
    irrelevant (inj₁  a) (inj₂ ¬a)  = ⊥-elim (¬a a)
    irrelevant (inj₂ ¬a) (inj₁  a)  = ⊥-elim (¬a a)
    irrelevant (inj₂ ¬a) (inj₂ ¬a′) =
      cong (inj₂ {A = A}) $ proj₁ $ ¬-propositional ext ¬a ¬a′

  -- Is-proposition is also closed under _Xor_ (assuming
  -- extensionality).

  Xor-closure-propositional :
     {a b} {A : Set a} {B : Set b} 
    Extensionality (a  b) (# 0) 
    Is-proposition A  Is-proposition B 
    Is-proposition (A Xor B)
  Xor-closure-propositional {ℓa} {ℓb} {A} {B} ext pA pB =
    _⇔_.from propositional⇔irrelevant irr
    where
    irr : (x y : A Xor B)  x  y
    irr (inj₁ (a , ¬b)) (inj₂ (¬a  , b))   = ⊥-elim (¬a a)
    irr (inj₂ (¬a , b)) (inj₁ (a   , ¬b))  = ⊥-elim (¬b b)
    irr (inj₁ (a , ¬b)) (inj₁ (a′  , ¬b′)) =
      cong₂  x y  inj₁ (x , y))
        (_⇔_.to propositional⇔irrelevant pA a a′)
        (lower-extensionality ℓa _ ext λ b  ⊥-elim (¬b b))
    irr (inj₂ (¬a , b)) (inj₂ (¬a′ , b′)) =
      cong₂  x y  inj₂ (x , y))
        (lower-extensionality ℓb _ ext λ a  ⊥-elim (¬a a))
        (_⇔_.to propositional⇔irrelevant pB b b′)

  -- However, H-level is not closed under _Xor_.

  ¬-Xor-closure-contractible :  {a b} 
    ¬ ({A : Set a} {B : Set b} 
       Contractible A  Contractible B  Contractible (A Xor B))
  ¬-Xor-closure-contractible closure
    with proj₁ $ closure (↑-closure 0 ⊤-contractible)
                         (↑-closure 0 ⊤-contractible)
  ... | inj₁ (_ , ¬⊤) = ¬⊤ _
  ... | inj₂ (¬⊤ , _) = ¬⊤ _

  -- Alternative definition of ⊎-closure (for Set₀).

  module Alternative-proof where

    -- Is-set is closed under _⊎_, by an argument similar to the one
    -- Hedberg used to prove that decidable equality implies
    -- uniqueness of identity proofs.

    ⊎-closure-set : {A B : Set} 
                    Is-set A  Is-set B  Is-set (A  B)
    ⊎-closure-set {A} {B} A-set B-set =
      _⇔_.from set⇔UIP (DUIP.constant⇒UIP c)
      where
      c : (x y : A  B)   λ (f : x  y  x  y)  DUIP.Constant f
      c (inj₁ x) (inj₁ y) = (cong inj₁  ⊎.cancel-inj₁ , λ p q  cong (cong inj₁) $ proj₁ $ A-set x y (⊎.cancel-inj₁ p) (⊎.cancel-inj₁ q))
      c (inj₂ x) (inj₂ y) = (cong inj₂  ⊎.cancel-inj₂ , λ p q  cong (cong inj₂) $ proj₁ $ B-set x y (⊎.cancel-inj₂ p) (⊎.cancel-inj₂ q))
      c (inj₁ x) (inj₂ y) = (⊥-elim  ⊎.inj₁≢inj₂       , λ _  ⊥-elim  ⊎.inj₁≢inj₂)
      c (inj₂ x) (inj₁ y) = (⊥-elim  ⊎.inj₁≢inj₂  sym , λ _  ⊥-elim  ⊎.inj₁≢inj₂  sym)

    -- H-level is closed under _⊎_ for other levels greater than or equal
    -- to 2 too.

    ⊎-closure′ :
       {A B : Set} n 
      H-level (2 + n) A  H-level (2 + n) B  H-level (2 + n) (A  B)
    ⊎-closure′         zero    = ⊎-closure-set
    ⊎-closure′ {A} {B} (suc n) = clos
      where
      clos : H-level (3 + n) A  H-level (3 + n) B  H-level (3 + n) (A  B)
      clos hA hB (inj₁ x) (inj₁ y) = respects-surjection (_↔_.surjection ≡↔inj₁≡inj₁) (2 + n) (hA x y)
      clos hA hB (inj₂ x) (inj₂ y) = respects-surjection (_↔_.surjection ≡↔inj₂≡inj₂) (2 + n) (hB x y)
      clos hA hB (inj₁ x) (inj₂ y) = ⊥-elim  ⊎.inj₁≢inj₂
      clos hA hB (inj₂ x) (inj₁ y) = ⊥-elim  ⊎.inj₁≢inj₂  sym