38 lines
1.1 KiB
Text
38 lines
1.1 KiB
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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-- TODO: take a look at the Coq tricks
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import hott.path hott.equiv
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open path
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set_option pp.universes true
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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.{l} [class] : Type.{l+3} :=
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mk : (Π (A : Type.{l+1}) (P : A → Type.{l+2}) (f g : Π x, P x), IsEquiv (@apD10 A P f g))
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→ funext.{l}
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namespace funext
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context
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universe l
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parameters [F : funext.{l}] {A : Type.{l+1}} {P : A → Type.{l+2}} (f g : Π x, P x)
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protected definition ap [instance] : IsEquiv (@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_forall : f ∼ g → f ≈ g :=
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@IsEquiv.inv _ _ (@apD10 A P f g) ap
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end
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definition path_forall2 [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_forall f g (λx, path_forall (f x) (g x) (E x))
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end funext
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