lean2/library/hott/funext.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: Jeremy Avigad
-- Ported from Coq HoTT
-- TODO: move this to an "axioms" folder
-- TODO: take a look at the Coq tricks
import .path .equiv
open path
-- Funext
-- ------
axiom funext {A : Type} {P : A → Type} (f g : Π x : A, P x) : IsEquiv (@apD10 A P f g)
definition path_forall {A : Type} {P : A → Type} (f g : Π x : A, P x) : f g → f ≈ g :=
(funext f g)⁻¹
definition path_forall2 {A B : Type} {P : A → B → Type} (f g : Π x y, P x y) :
(Πx y, f x y ≈ g x y) → f ≈ g :=
λE, path_forall f g (λx, path_forall (f x) (g x) (E x))