61901cff81
also add more definitions in types.pi, types.path, algebra.precategory the (pre)category library still needs cleanup authors of this commit: @avigad, @javra, @fpvandoorn
129 lines
4.9 KiB
Text
129 lines
4.9 KiB
Text
-- Copyright (c) 2014 Jakob von Raumer. All rights reserved.
|
||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||
-- Author: Jakob von Raumer
|
||
-- Ported from Coq HoTT
|
||
prelude
|
||
import ..equiv ..datatypes ..types.prod
|
||
import .funext_varieties .ua .funext
|
||
|
||
open eq function prod is_trunc sigma equiv is_equiv unit
|
||
|
||
context
|
||
universe variables l
|
||
|
||
private theorem ua_isequiv_postcompose {A B : Type.{l}} {C : Type}
|
||
{w : A → B} [H0 : is_equiv w] : is_equiv (@compose C A B w) :=
|
||
let w' := equiv.mk w H0 in
|
||
let eqinv : A = B := ((@is_equiv.inv _ _ _ (univalence A B)) w') in
|
||
let eq' := equiv_of_eq eqinv in
|
||
is_equiv.adjointify (@compose C A B w)
|
||
(@compose C B A (is_equiv.inv w))
|
||
(λ (x : C → B),
|
||
have eqretr : eq' = w',
|
||
from (@retr _ _ (@equiv_of_eq A B) (univalence A B) w'),
|
||
have invs_eq : (to_fun eq')⁻¹ = (to_fun w')⁻¹,
|
||
from inv_eq eq' w' eqretr,
|
||
have eqfin : (to_fun eq') ∘ ((to_fun eq')⁻¹ ∘ x) = x,
|
||
from (λ p,
|
||
(@eq.rec_on Type.{l} A
|
||
(λ B' p', Π (x' : C → B'), (to_fun (equiv_of_eq p'))
|
||
∘ ((to_fun (equiv_of_eq p'))⁻¹ ∘ x') = x')
|
||
B p (λ x', idp))
|
||
) eqinv x,
|
||
have eqfin' : (to_fun w') ∘ ((to_fun eq')⁻¹ ∘ x) = x,
|
||
from eqretr ▹ eqfin,
|
||
have eqfin'' : (to_fun w') ∘ ((to_fun w')⁻¹ ∘ x) = x,
|
||
from invs_eq ▹ eqfin',
|
||
eqfin''
|
||
)
|
||
(λ (x : C → A),
|
||
have eqretr : eq' = w',
|
||
from (@retr _ _ (@equiv_of_eq A B) (univalence A B) w'),
|
||
have invs_eq : (to_fun eq')⁻¹ = (to_fun w')⁻¹,
|
||
from inv_eq eq' w' eqretr,
|
||
have eqfin : (to_fun eq')⁻¹ ∘ ((to_fun eq') ∘ x) = x,
|
||
from (λ p, eq.rec_on p idp) eqinv,
|
||
have eqfin' : (to_fun eq')⁻¹ ∘ ((to_fun w') ∘ x) = x,
|
||
from eqretr ▹ eqfin,
|
||
have eqfin'' : (to_fun w')⁻¹ ∘ ((to_fun w') ∘ x) = x,
|
||
from invs_eq ▹ eqfin',
|
||
eqfin''
|
||
)
|
||
|
||
-- We are ready to prove functional extensionality,
|
||
-- starting with the naive non-dependent version.
|
||
private definition diagonal [reducible] (B : Type) : Type
|
||
:= Σ xy : B × B, pr₁ xy = pr₂ xy
|
||
|
||
private definition isequiv_src_compose {A B : Type}
|
||
: @is_equiv (A → diagonal B)
|
||
(A → B)
|
||
(compose (pr₁ ∘ pr1)) :=
|
||
@ua_isequiv_postcompose _ _ _ (pr₁ ∘ pr1)
|
||
(is_equiv.adjointify (pr₁ ∘ pr1)
|
||
(λ x, sigma.mk (x , x) idp) (λx, idp)
|
||
(λ x, sigma.rec_on x
|
||
(λ xy, prod.rec_on xy
|
||
(λ b c p, eq.rec_on p idp))))
|
||
|
||
private definition isequiv_tgt_compose {A B : Type}
|
||
: @is_equiv (A → diagonal B)
|
||
(A → B)
|
||
(compose (pr₂ ∘ pr1)) :=
|
||
@ua_isequiv_postcompose _ _ _ (pr2 ∘ pr1)
|
||
(is_equiv.adjointify (pr2 ∘ pr1)
|
||
(λ x, sigma.mk (x , x) idp) (λx, idp)
|
||
(λ x, sigma.rec_on x
|
||
(λ xy, prod.rec_on xy
|
||
(λ b c p, eq.rec_on p idp))))
|
||
|
||
theorem nondep_funext_from_ua {A : Type} {B : Type}
|
||
: Π {f g : A → B}, f ∼ g → f = g :=
|
||
(λ (f g : A → B) (p : f ∼ g),
|
||
let d := λ (x : A), sigma.mk (f x , f x) idp in
|
||
let e := λ (x : A), sigma.mk (f x , g x) (p x) in
|
||
let precomp1 := compose (pr₁ ∘ pr1) in
|
||
have equiv1 [visible] : is_equiv precomp1,
|
||
from @isequiv_src_compose A B,
|
||
have equiv2 [visible] : Π x y, is_equiv (ap precomp1),
|
||
from is_equiv.is_equiv_ap precomp1,
|
||
have H' : Π (x y : A → diagonal B),
|
||
pr₁ ∘ pr1 ∘ x = pr₁ ∘ pr1 ∘ y → x = y,
|
||
from (λ x y, is_equiv.inv (ap precomp1)),
|
||
have eq2 : pr₁ ∘ pr1 ∘ d = pr₁ ∘ pr1 ∘ e,
|
||
from idp,
|
||
have eq0 : d = e,
|
||
from H' d e eq2,
|
||
have eq1 : (pr₂ ∘ pr1) ∘ d = (pr₂ ∘ pr1) ∘ e,
|
||
from ap _ eq0,
|
||
eq1
|
||
)
|
||
|
||
end
|
||
|
||
-- Now we use this to prove weak funext, which as we know
|
||
-- implies (with dependent eta) also the strong dependent funext.
|
||
theorem weak_funext_of_ua : weak_funext :=
|
||
(λ (A : Type) (P : A → Type) allcontr,
|
||
let U := (λ (x : A), unit) in
|
||
have pequiv : Π (x : A), P x ≃ U x,
|
||
from (λ x, @equiv_unit_of_is_contr (P x) (allcontr x)),
|
||
have psim : Π (x : A), P x = U x,
|
||
from (λ x, @is_equiv.inv _ _
|
||
equiv_of_eq (univalence _ _) (pequiv x)),
|
||
have p : P = U,
|
||
from @nondep_funext_from_ua A Type P U psim,
|
||
have tU' : is_contr (A → unit),
|
||
from is_contr.mk (λ x, ⋆)
|
||
(λ f, @nondep_funext_from_ua A unit (λ x, ⋆) f
|
||
(λ x, unit.rec_on (f x) idp)),
|
||
have tU : is_contr (Π x, U x),
|
||
from tU',
|
||
have tlast : is_contr (Πx, P x),
|
||
from eq.transport _ (p⁻¹) tU,
|
||
tlast
|
||
)
|
||
|
||
-- In the following we will proof function extensionality using the univalence axiom
|
||
definition funext_of_ua [instance] : funext :=
|
||
funext_of_weak_funext (@weak_funext_of_ua)
|