d8a616fa70
I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
33 lines
1.3 KiB
Text
33 lines
1.3 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.inhabited logic.cast
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open inhabited
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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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-- TODO: generalize to eq_rec
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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 [visible] : inhabited (Π x, B x), from inhabited.mk 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 hcongr_fun {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 [visible] : inhabited (Π x, B x), from inhabited.mk f,
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have Hb : B = B', from piext (heq.type_eq H),
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hcongr_fun' 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, hcongr_fun 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 heq.subst Haa' H1,
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H2 B B' f f' Hff'
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