lean2/library/standard/piext.lean

-- 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'