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

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

module Function where

open import Level
open import Strict

private
  variable
    a b c d e : Level
    A : Set a
    B : Set b
    C : Set c
    D : Set d
    E : Set e

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

Fun₁ : Set a  Set a
Fun₁ A = A  A

Fun₂ : Set a  Set a
Fun₂ A = A  A  A

------------------------------------------------------------------------
-- Some simple functions

id : A  A
id x = x

const : A  B  A
const x = λ _  x

------------------------------------------------------------------------
-- Operations on dependent functions

-- These are functions whose output has a type that depends on the
-- value of the input to the function.

infixr 9 _∘_
infixl 8 _ˢ_
infixl 0 _|>_
infix  0 case_return_of_
infixr -1 _$_ _$!_

-- Composition

_∘_ :  {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)

-- Flipping order of arguments

flip :  {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

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

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

-- Strict (call-by-value) application

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

-- Flipped application (aka pipe-forward)

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

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

_ˢ_ :  {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)

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

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

------------------------------------------------------------------------
-- Non-dependent versions of dependent operations

-- Any of the above operations for dependent functions will also work
-- for non-dependent functions but sometimes Agda has difficulty
-- inferring the non-dependency. Primed (′ = \prime) versions of the
-- operations are therefore provided below that sometimes have better
-- inference properties.

infixr 9 _∘′_
infixl 0 _|>′_
infix  0 case_of_
infixr -1 _$′_ _$!′_

-- Composition

_∘′_ : (B  C)  (A  B)  (A  C)
f ∘′ g = _∘_ f g

-- Application

_$′_ : (A  B)  (A  B)
_$′_ = _$_

-- Strict (call-by-value) application

_$!′_ : (A  B)  (A  B)
_$!′_ = _$!_

-- Flipped application (aka pipe-forward)

_|>′_ : A  (A  B)  B
_|>′_ = _|>_

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

case_of_ : A  (A  B)  B
case x of f = case x return _ of f

------------------------------------------------------------------------
-- Operations that are only defined for non-dependent functions

infixr 0 _-[_]-_
infixl 1 _on_
infixl 1 _⟨_⟩_
infixl 0 _∋_

-- Binary application

_⟨_⟩_ : A  (A  B  C)  B  C
x  f  y = f x y

-- Composition of a binary function with a unary function

_on_ : (B  B  C)  (A  B)  (A  A  C)
_*_ on f = λ x y  f x * f y

-- Composition of three binary functions

_-[_]-_ : (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 : 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 : Set a}  A  Set a
typeOf {A = A} _ = A