------------------------------------------------------------------------
-- Simple combinators working solely on and with functions
------------------------------------------------------------------------

{-# OPTIONS --universe-polymorphism #-}

module Function where

open import Level

infixr 9 _∘_ _∘′_
infixl 1 _on_
infixl 1 _⟨_⟩_
infixr 0 _-[_]-_ _$_
infix  0 type-signature

------------------------------------------------------------------------
-- Types

-- Unary functions on a given set.

Fun₁ :  {i}  Set i  Set i
Fun₁ A = A  A

-- Binary functions on a given set.

Fun₂ :  {i}  Set i  Set i
Fun₂ A = A  A  A

------------------------------------------------------------------------
-- Functions

_∘_ :  {a b c}
        {A : Set a} {B : A  Set b} {C : {x : A}  B x  Set c} 
      (∀ {x} (y : B x)  C y)  (g : (x : A)  B x) 
      ((x : A)  C (g x))
f  g = λ x  f (g x)

_∘′_ :  {a b c} {A : Set a} {B : Set b} {C : Set c} 
       (B  C)  (A  B)  (A  C)
f ∘′ g = _∘_ f g

id :  {a} {A : Set a}  A  A
id x = x

const :  {a b} {A : Set a} {B : Set b}  A  B  A
const x = λ _  x

flip :  {a b c} {A : Set a} {B : Set b} {C : A  B  Set c} 
       ((x : A) (y : B)  C x y)  ((y : B) (x : A)  C x y)
flip f = λ x y  f y x

-- Note that _$_ is right associative, like in Haskell. If you want a
-- left associative infix application operator, use
-- Category.Functor._<$>_, available from
-- Category.Monad.Identity.IdentityMonad.

_$_ :  {a b} {A : Set a} {B : A  Set b} 
      ((x : A)  B x)  ((x : A)  B x)
f $ x = f x

_⟨_⟩_ :  {a b c} {A : Set a} {B : Set b} {C : Set c} 
        A  (A  B  C)  B  C
x  f  y = f x y

_on_ :  {a b c} {A : Set a} {B : Set b} {C : Set c} 
       (B  B  C)  (A  B)  (A  A  C)
_*_ on f = λ x y  f x * f y

_-[_]-_ :  {a b c d e} {A : Set a} {B : Set b} {C : Set c}
            {D : Set d} {E : Set e} 
          (A  B  C)  (C  D  E)  (A  B  D)  (A  B  E)
f -[ _*_ ]- g = λ x y  f x y * g x y

-- In Agda you cannot annotate every subexpression with a type
-- signature. This function can be used instead.

type-signature :  {a} (A : Set a)  A  A
type-signature A x = x

syntax type-signature A x = x ∶ A