31 lines
1.5 KiB
Text
31 lines
1.5 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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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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opaque_hint (hiding transport)
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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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theorem transport_trans {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p, calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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