lean2/library/logic/axioms/piext.lean
Floris van Doorn d8a616fa70 refactor(library): major changes in the library
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
2014-11-03 18:45:12 -08:00

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
import logic.inhabited logic.cast
open inhabited
-- Pi extensionality
axiom piext {A : Type} {B B' : A → Type} [H : inhabited (Π x, B x)] :
(Π x, B x) = (Π x, B' x) → B = B'
-- TODO: generalize to eq_rec
theorem cast_app {A : Type} {B B' : A → Type} (H : (Π x, B x) = (Π x, B' x)) (f : Π x, B x)
(a : A) : cast H f a == f a :=
have Hi [visible] : inhabited (Π x, B x), from inhabited.mk f,
have Hb : B = B', from piext H,
cast_app' Hb f a
theorem hcongr_fun {A : Type} {B B' : A → Type} {f : Π x, B x} {f' : Π x, B' x} (a : A)
(H : f == f') : f a == f' a :=
have Hi [visible] : inhabited (Π x, B x), from inhabited.mk f,
have Hb : B = B', from piext (heq.type_eq H),
hcongr_fun' a H Hb
theorem hcongr {A A' : Type} {B : A → Type} {B' : A' → Type}
{f : Π x, B x} {f' : Π x, B' x} {a : A} {a' : A'}
(Hff' : f == f') (Haa' : a == a') : f a == f' a' :=
have H1 : ∀ (B B' : A → Type) (f : Π x, B x) (f' : Π x, B' x), f == f' → f a == f' a, from
take B B' f f' e, hcongr_fun a e,
have H2 : ∀ (B : A → Type) (B' : A' → Type) (f : Π x, B x) (f' : Π x, B' x),
f == f' → f a == f' a', from heq.subst Haa' H1,
H2 B B' f f' Hff'