lean2/library/algebra/function.lean

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
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
-- Yet another trick to anotate an expression with a type
definition is_typeof (A : Type) (a : A) : A :=
a
precedence `∘`:60
precedence `∘'`:60
precedence `on`:1
precedence `$`:1
infixr ∘ := compose
infixr ∘' := dcompose
infixl on := on_fun
notation `typeof` t `:` T := is_typeof T t
infixr $ := app
notation f `-[` op `]-` g := combine f op g
-- Trick for using any binary function as infix operator
notation a `⟨` f `⟩` b := f a b
end function