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