32 lines
1.5 KiB
Text
32 lines
1.5 KiB
Text
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-- 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.classes.inhabited logic.connectives.cast
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-- Pi extensionality
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axiom piext {A : Type} {B B' : A → Type} {H : inhabited (Π x, B x)} :
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(Π x, B x) = (Π x, B' x) → B = B'
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theorem cast_app {A : Type} {B B' : A → Type} (H : (Π x, B x) = (Π x, B' x)) (f : Π x, B x)
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(a : A) : cast H f a == f a :=
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have Hi [fact] : inhabited (Π x, B x), from inhabited_intro f,
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have Hb : B = B', from piext H,
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cast_app' Hb f a
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theorem hcongr1 {A : Type} {B B' : A → Type} {f : Π x, B x} {f' : Π x, B' x} (a : A)
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(H : f == f') : f a == f' a :=
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have Hi [fact] : inhabited (Π x, B x), from inhabited_intro f,
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have Hb : B = B', from piext (type_eq H),
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hcongr1' a H Hb
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theorem hcongr {A A' : Type} {B : A → Type} {B' : A' → Type}
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{f : Π x, B x} {f' : Π x, B' x} {a : A} {a' : A'}
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(Hff' : f == f') (Haa' : a == a') : f a == f' a' :=
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have H1 : ∀ (B B' : A → Type) (f : Π x, B x) (f' : Π x, B' x), f == f' → f a == f' a, from
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take B B' f f' e, hcongr1 a e,
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have H2 : ∀ (B : A → Type) (B' : A' → Type) (f : Π x, B x) (f' : Π x, B' x),
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f == f' → f a == f' a', from hsubst Haa' H1,
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H2 B B' f f' Hff'
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