34 lines
1,006 B
Text
34 lines
1,006 B
Text
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||
-- Author: Jeremy Avigad, Jakob von Raumer
|
||
-- Ported from Coq HoTT
|
||
prelude
|
||
import ..path ..equiv
|
||
open eq
|
||
|
||
-- Funext
|
||
-- ------
|
||
|
||
-- Define function extensionality as a type class
|
||
inductive funext [class] : Type :=
|
||
mk : (Π (A : Type) (P : A → Type ) (f g : Π x, P x), is_equiv (@apD10 A P f g))
|
||
→ funext
|
||
|
||
namespace funext
|
||
|
||
universe variables l k
|
||
variables [F : funext.{l k}] {A : Type.{l}} {P : A → Type.{k}}
|
||
|
||
include F
|
||
protected definition ap [instance] (f g : Π x, P x) : is_equiv (@apD10 A P f g) :=
|
||
funext.rec_on F (λ(H : Π A P f g, _), !H)
|
||
|
||
definition path_pi {f g : Π x, P x} : f ∼ g → f = g :=
|
||
is_equiv.inv (@apD10 A P f g)
|
||
|
||
omit F
|
||
definition path_pi2 [F : funext] {A B : Type} {P : A → B → Type}
|
||
(f g : Πx y, P x y) : (Πx y, f x y = g x y) → f = g :=
|
||
λ E, path_pi (λx, path_pi (E x))
|
||
|
||
end funext
|