49 lines
1.3 KiB
Text
49 lines
1.3 KiB
Text
/-
|
|
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
|
|
Module: algebra.function
|
|
Author: Leonardo de Moura
|
|
|
|
General operations on functions.
|
|
-/
|
|
namespace function
|
|
|
|
variables {A : Type} {B : Type} {C : Type} {D : Type} {E : Type}
|
|
|
|
definition compose [reducible] (f : B → C) (g : A → B) : A → C :=
|
|
λx, f (g x)
|
|
|
|
definition id [reducible] (a : A) : A :=
|
|
a
|
|
|
|
definition on_fun (f : B → B → C) (g : A → B) : A → A → C :=
|
|
λx y, f (g x) (g y)
|
|
|
|
definition combine (f : A → B → C) (op : C → D → E) (g : A → B → D) : A → B → E :=
|
|
λx y, op (f x y) (g x y)
|
|
|
|
definition const {A : Type} (B : Type) (a : A) : B → A :=
|
|
λx, a
|
|
|
|
definition dcompose {A : Type} {B : A → Type} {C : Π {x : A}, B x → Type}
|
|
(f : Π {x : A} (y : B x), C y) (g : Πx, B x) : Πx, C (g x) :=
|
|
λx, f (g x)
|
|
|
|
definition flip {A : Type} {B : Type} {C : A → B → Type} (f : Πx y, C x y) : Πy x, C x y :=
|
|
λy x, f x y
|
|
|
|
definition app {A : Type} {B : A → Type} (f : Πx, B x) (x : A) : B x :=
|
|
f x
|
|
|
|
precedence `∘'`:60
|
|
precedence `on`:1
|
|
precedence `$`:1
|
|
|
|
infixr ∘ := compose
|
|
infixr ∘' := dcompose
|
|
infixl on := on_fun
|
|
infixr $ := app
|
|
notation f `-[` op `]-` g := combine f op g
|
|
|
|
end function
|