lean2/library/standard/piext.lean

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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 cast
-- 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'
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 [fact] : inhabited (Π x, B x), from inhabited_intro f,
have Hb : B = B', from piext H,
have H1 : ∀ (H : (Π x, B x) = (Π x, B x)), cast H f a == f a, from
assume H, eq_to_heq (congr1 (cast_eq H f) a),
have H2 : ∀ (H : (Π x, B x) = (Π x, B' x)), cast H f a == f a, from
subst Hb H1,
H2 H
theorem cast_pull {A : Type} {B B' : A → Type} (f : Π x, B x) (a : A) (Hb : (Π x, B x) = (Π x, B' x)) (Hba : (B a) = (B' a)) :
cast Hb f a = cast Hba (f a)
:= heq_to_eq (calc cast Hb f a == f a : cast_app Hb f a
... == cast Hba (f a) : hsymm (cast_heq Hba (f a)))
theorem hcongr1 {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
:= heq_elim H (λ (Ht : (Π x, B x) = (Π x, B' x)) (Hw : cast Ht f = f'),
calc f a == cast Ht f a : hsymm (cast_app Ht f a)
... = f' a : congr1 Hw a)
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, hcongr1 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
hsubst Haa' H1,
H2 B B' f f' Hff'