25 lines
1.1 KiB
Text
25 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: Leonardo de Moura
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import logic
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open eq
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definition refl := @eq.refl
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definition transport {A : Type} {a b : A} {P : A → Type} (p : a = b) (H : P a) : P b
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:= eq.rec H p
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theorem transport_refl {A : Type} {a : A} {P : A → Type} (H : P a) : transport (refl a) H = H
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:= refl H
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attribute transport [irreducible]
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theorem transport_proof_irrel {A : Type} {a b : A} {P : A → Type} (p1 p2 : a = b) (H : P a) : transport p1 H = transport p2 H
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:= refl (transport p1 H)
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theorem transport_eq {A : Type} {a : A} {P : A → Type} (p : a = a) (H : P a) : transport p H = H
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:= calc transport p H = transport (refl a) H : transport_proof_irrel p (refl a) H
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... = H : transport_refl H
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theorem dcongr {A : Type} {B : A → Type} {a b : A} (f : Π x, B x) (p : a = b) : transport p (f a) = f b
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:= have H1 : ∀ p1 : a = a, transport p1 (f a) = f a, from
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assume p1 : a = a, transport_eq p1 (f a),
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eq.rec H1 p p
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