{-
  Definitions for functions
-}
{-# OPTIONS --safe #-}
module Cubical.Foundations.Function where

open import Cubical.Foundations.Prelude

private
  variable
     ℓ' ℓ'' : Level
    A A' A'' : Type 
    B : A  Type 
    C : (a : A)  B a  Type 
    D : (a : A) (b : B a)  C a b  Type 
    E : (x : A)  (y : B x)  (z : C x y)  (w : D x y z)  Type 
    F : (x : A)  (y : B x)  (z : C x y)  (w : D x y z)  (u : E x y z w)  Type 

-- The identity function
idfun : (A : Type )  A  A
idfun _ x = x

infixr -1 _$_

_$_ :  { ℓ'} {A : Type } {B : A  Type ℓ'}  ((a : A)  B a)  (a : A)  B a
f $ a = f a
{-# INLINE _$_ #-}

infixr 9 _∘_ _∘S_

_∘_ : (g : {a : A}  (b : B a)  C a b)  (f : (a : A)  B a)  (a : A)  C a (f a)
g  f = λ x  g (f x)
{-# INLINE _∘_ #-}

_∘S_ : (A'  A'')  (A  A')  A  A''
g ∘S f = λ x  g (f x)
{-# INLINE _∘S_ #-}

∘-assoc : (h : {a : A} {b : B a}  (c : C a b)  D a b c)
          (g : {a : A}  (b : B a)  C a b)
          (f : (a : A)  B a)
         (h  g)  f  h  (g  f)
∘-assoc h g f i x = h (g (f x))

∘-idˡ : (f : (a : A)  B a)  f  idfun A  f
∘-idˡ f i x = f x

∘-idʳ : (f : (a : A)  B a)   {a}  idfun (B a))  f  f
∘-idʳ f i x = f x

flip : {B : Type ℓ'} {C : A  B  Type ℓ''} 
       ((x : A) (y : B)  C x y)  ((y : B) (x : A)  C x y)
flip f y x = f x y
{-# INLINE flip #-}

const : {B : Type }  A  B  A
const x = λ _  x
{-# INLINE const #-}

case_of_ :  { ℓ'} {A : Type } {B : A  Type ℓ'}  (x : A)  (∀ x  B x)  B x
case x of f = f x
{-# INLINE case_of_ #-}

case_return_of_ :  { ℓ'} {A : Type } (x : A) (B : A  Type ℓ')  (∀ x  B x)  B x
case x return P of f = f x
{-# INLINE case_return_of_ #-}

uncurry : ((x : A)  (y : B x)  C x y)  (p : Σ A B)  C (fst p) (snd p)
uncurry f (x , y) = f x y

uncurry2 : ((x : A)  (y : B x)  (z : C x y)  D x y z)
          (p : Σ A  x  Σ (B x) (C x)))  D (p .fst) (p .snd .fst) (p .snd .snd)
uncurry2 f (x , y , z) = f x y z

uncurry3 : ((x : A)  (y : B x)  (z : C x y)  (w : D x y z)  E x y z w)
          (p : Σ A  x  Σ (B x)  y  Σ (C x y) (D x y))))
          E (p .fst) (p .snd .fst) (p .snd .snd .fst) (p .snd .snd .snd)
uncurry3 f (x , y , z , w) = f x y z w

uncurry4 : ((x : A)  (y : B x)  (z : C x y)  (w : D x y z)  (u : E x y z w)  F x y z w u)
          (p : Σ A  x  Σ (B x)  y  Σ (C x y)  z  Σ (D x y z) (E x y z)))))
          F (p .fst) (p .snd .fst) (p .snd .snd .fst) (p .snd .snd .snd .fst) (p .snd .snd .snd .snd)
uncurry4 f (x , y , z , w , u) = f x y z w u


curry : ((p : Σ A B)  C (fst p) (snd p))  (x : A)  (y : B x)  C x y
curry f x y = f (x , y)

∘diag : {B : (x y : A)  Type }  (∀ x y  B x y)   x  B x x
∘diag f x = f x x

module _ { ℓ'} {A : Type } {B : Type ℓ'} where
  -- Notions of 'coherently constant' functions for low dimensions.
  -- These are the properties of functions necessary to e.g. eliminate
  -- from the propositional truncation.

  -- 2-Constant functions are coherently constant if B is a set.
  2-Constant : (A  B)  Type _
  2-Constant f =  x y  f x  f y

  2-Constant-isProp : isSet B  (f : A  B)  isProp (2-Constant f)
  2-Constant-isProp Bset f link1 link2 i x y j
    = Bset (f x) (f y) (link1 x y) (link2 x y) i j

  -- 3-Constant functions are coherently constant if B is a groupoid.
  record 3-Constant (f : A  B) : Type (ℓ-max  ℓ') where
    field
      link : 2-Constant f
      coh₁ :  x y z  Square (link x y) (link x z) refl (link y z)

    coh₂ :  x y z  Square (link x z) (link y z) (link x y) refl
    coh₂ x y z i j
      = hcomp  k  λ
              { (j = i0)  link x y i
              ; (i = i0)  link x z (j  k)
              ; (j = i1)  link x z (i  k)
              ; (i = i1)  link y z j
              })
          (coh₁ x y z j i)

    link≡refl :  x  refl  link x x
    link≡refl x i j
      = hcomp  k  λ
              { (i = i0)  link x x (j  ~ k)
              ; (i = i1)  link x x j
              ; (j = i0)  f x
              ; (j = i1)  link x x (~ i  ~ k)
              })
          (coh₁ x x x (~ i) j)

    downleft :  x y  Square (link x y) refl refl (link y x)
    downleft x y i j
      = hcomp  k  λ
              { (i = i0)  link x y j
              ; (i = i1)  link≡refl x (~ k) j
              ; (j = i0)  f x
              ; (j = i1)  link y x i
              })
          (coh₁ x y x i j)

    link≡symlink :  x y  link x y  sym (link y x)
    link≡symlink x y i j
      = hcomp  k  λ
              { (i = i0)  link x y j
              ; (i = i1)  link y x (~ j  ~ k)
              ; (j = i0)  f x
              ; (j = i1)  link y x (i  ~ k)
              })
          (downleft x y i j)

homotopyNatural : {B : Type ℓ'} {f g : A  B} (H :  a  f a  g a) {x y : A} (p : x  y) 
                  H x  cong g p  cong f p  H y
homotopyNatural {f = f} {g = g} H {x} {y} p i j =
    hcomp  k  λ { (i = i0)  compPath-filler (H x) (cong g p) k j
                   ; (i = i1)  compPath-filler' (cong f p) (H y) k j
                   ; (j = i0)  cong f p (i  ~ k)
                   ; (j = i1)  cong g p (i  k) })
          (H (p i) j)

homotopySymInv : {f : A  A} (H :  a  f a  a) (a : A)
                Path (f a  f a)  i  H (H a (~ i)) i) refl
homotopySymInv {f = f} H a j i =
  hcomp  k  λ { (i = i0)  f a
                 ; (i = i1)  H a (j  ~ k)
                 ; (j = i0)  H (H a (~ i)) i
                 ; (j = i1)  H a (i  ~ k)})
        (H (H a (~ i  j)) i)