------------------------------------------------------------------------
-- The Agda standard library
--
-- Simple combinators working solely on and with functions
------------------------------------------------------------------------

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

module Function where

open import Level
open import Strict

infixr 9 _∘_ _∘′_
infixl 8 _ˢ_
infixl 1 _on_
infixl 1 _⟨_⟩_
infixr -1 _$_ _$′_ _$!_ _$!′_
infixr 0 _-[_]-_
infixl 0 _|>_ _|>′_ _∋_

------------------------------------------------------------------------
-- 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

-- The S combinator. (Written infix as in Conor McBride's paper
-- "Outrageous but Meaningful Coincidences: Dependent type-safe syntax
-- and evaluation".)

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

-- 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

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

-- Strict (call-by-value) application

_$!_ :  {a b} {A : Set a} {B : A  Set b} 
       ((x : A)  B x)  ((x : A)  B x)
_$!_ = flip force

_$!′_ :  {a b} {A : Set a} {B : Set b} 
        (A  B)  (A  B)
_$!′_ = _$!_

-- flipped application aka pipe-forward

_|>_ :  {a b} {A : Set a} {B : A  Set b} 
       (a : A)  (∀ a  B a)  B a
_|>_ = flip _$_


_|>′_ :  {a b} {A : Set a} {B : Set b} 
        A  (A  B)  B
_|>′_ = _|>_


_⟨_⟩_ :  {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.

_∋_ :  {a} (A : Set a)  A  A
A  x = x

-- Conversely it is sometimes useful to be able to extract the
-- type of a given expression:

typeOf :  {a} {A : Set a}  A  Set a
typeOf {A = A} _ = A

-- Case expressions (to be used with pattern-matching lambdas, see
-- README.Case).

infix 0 case_return_of_ case_of_

case_return_of_ :
   {a b} {A : Set a}
  (x : A) (B : A  Set b)  ((x : A)  B x)  B x
case x return B of f = f x

case_of_ :  {a b} {A : Set a} {B : Set b}  A  (A  B)  B
case x of f = case x return _ of f