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

-- The contents of this file can be accessed from Function.

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

module Function.Base 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

------------------------------------------------------------------------
-- 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 _$_ -- The S combinator - written infix as in Conor McBride's paper -- "Outrageous but Meaningful Coincidences: Dependent type-safe syntax -- and evaluation". _ˢ_ : ∀ {A : Set a} {B : A → Set b} {C : (x : A) → B x → Set c} → ((x : A) (y : B x) → C x y) → (g : (x : A) → B x) → ((x : A) → C x (g x)) f ˢ g = λ x → f x (g x) -- Converting between implicit and explicit function spaces. _$- : ∀ {A : Set a} {B : A → Set b} → ((x : A) → B x) → ({x : A} → B x)
f $- = f _ λ- : ∀ {A : Set a} {B : A → Set b} → ({x : A} → B x) → ((x : A) → B x) λ- f = λ x → f -- Case expressions (to be used with pattern-matching lambdas, see -- README.Case). case_return_of_ : ∀ {A : Set a} (x : A) (B : A → Set b) → ((x : A) → B x) → B x case x return B of f = f x ------------------------------------------------------------------------ -- Non-dependent versions of dependent operations -- Any of the above operations for dependent functions will also work -- for non-dependent functions but sometimes Agda has difficulty -- inferring the non-dependency. Primed (′ = \prime) versions of the -- operations are therefore provided below that sometimes have better -- inference properties. infixr 9 _∘′_ infixl 0 _|>′_ infix 0 case_of_ infixr -1 _$′_ _$!′_ -- Composition _∘′_ : (B → C) → (A → B) → (A → C) f ∘′ g = _∘_ f g -- Application _$′_ : (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

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

-- Construct an element of the given type by instance search.

it : {A : Set a} → {{A}} → A
it {{x}} = x