149 lines
5.7 KiB
Text
149 lines
5.7 KiB
Text
-- Copyright (c) 2014 Jakob von Raumer. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Jakob von Raumer
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-- Ported from Coq HoTT
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import hott.equiv hott.funext_varieties
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import data.prod data.sigma data.unit
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open path function prod sigma truncation Equiv IsEquiv unit
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set_option pp.universes true
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definition isequiv_path {A B : Type} (H : A ≈ B) :=
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(@IsEquiv.transport Type (λX, X) A B H)
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definition equiv_path {A B : Type} (H : A ≈ B) : A ≃ B :=
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Equiv.mk _ (isequiv_path H)
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-- First, define an axiom free variant of Univalence
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definition ua_type := Π (A B : Type), IsEquiv (@equiv_path A B)
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context
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universe variables l k
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parameter {ua : ua_type.{l+1}}
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protected theorem ua_isequiv_postcompose {A B : Type.{l+1}} {C : Type}
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{w : A → B} {H0 : IsEquiv w} : IsEquiv (@compose C A B w) :=
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let w' := Equiv.mk w H0 in
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let eqinv : A ≈ B := (equiv_path⁻¹ w') in
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let eq' := equiv_path eqinv in
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IsEquiv.adjointify (@compose C A B w)
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(@compose C B A (IsEquiv.inv w))
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(λ (x : C → B),
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have eqretr : eq' ≈ w',
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from (@retr _ _ (@equiv_path A B) (ua A B) w'),
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have invs_eq : (equiv_fun eq')⁻¹ ≈ (equiv_fun w')⁻¹,
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from inv_eq eq' w' eqretr,
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have eqfin : (equiv_fun eq') ∘ ((equiv_fun eq')⁻¹ ∘ x) ≈ x,
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from (λ p,
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(@path.rec_on Type.{l+1} A
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(λ B' p', Π (x' : C → B'), (equiv_fun (equiv_path p'))
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∘ ((equiv_fun (equiv_path p'))⁻¹ ∘ x') ≈ x')
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B p (λ x', idp))
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) eqinv x,
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have eqfin' : (equiv_fun w') ∘ ((equiv_fun eq')⁻¹ ∘ x) ≈ x,
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from eqretr ▹ eqfin,
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have eqfin'' : (equiv_fun w') ∘ ((equiv_fun w')⁻¹ ∘ x) ≈ x,
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from invs_eq ▹ eqfin',
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eqfin''
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)
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(λ (x : C → A),
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have eqretr : eq' ≈ w',
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from (@retr _ _ (@equiv_path A B) (ua A B) w'),
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have invs_eq : (equiv_fun eq')⁻¹ ≈ (equiv_fun w')⁻¹,
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from inv_eq eq' w' eqretr,
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have eqfin : (equiv_fun eq')⁻¹ ∘ ((equiv_fun eq') ∘ x) ≈ x,
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from (λ p, path.rec_on p idp) eqinv,
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have eqfin' : (equiv_fun eq')⁻¹ ∘ ((equiv_fun w') ∘ x) ≈ x,
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from eqretr ▹ eqfin,
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have eqfin'' : (equiv_fun w')⁻¹ ∘ ((equiv_fun w') ∘ x) ≈ x,
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from invs_eq ▹ eqfin',
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eqfin''
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)
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-- We are ready to prove functional extensionality,
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-- starting with the naive non-dependent version.
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protected definition diagonal [reducible] (B : Type) : Type
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:= Σ xy : B × B, pr₁ xy ≈ pr₂ xy
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protected definition isequiv_src_compose {A B : Type}
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: @IsEquiv (A → diagonal B)
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(A → B)
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(compose (pr₁ ∘ dpr1)) :=
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@ua_isequiv_postcompose _ _ _ (pr₁ ∘ dpr1)
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(IsEquiv.adjointify (pr₁ ∘ dpr1)
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(λ x, dpair (x , x) idp) (λx, idp)
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(λ x, sigma.rec_on x
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(λ xy, prod.rec_on xy
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(λ b c p, path.rec_on p idp))))
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protected definition isequiv_tgt_compose {A B : Type}
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: @IsEquiv (A → diagonal B)
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(A → B)
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(compose (pr₂ ∘ dpr1)) :=
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@ua_isequiv_postcompose _ _ _ (pr2 ∘ dpr1)
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(IsEquiv.adjointify (pr2 ∘ dpr1)
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(λ x, dpair (x , x) idp) (λx, idp)
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(λ x, sigma.rec_on x
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(λ xy, prod.rec_on xy
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(λ b c p, path.rec_on p idp))))
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theorem ua_implies_funext_nondep {A : Type} {B : Type.{l+1}}
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: Π {f g : A → B}, f ∼ g → f ≈ g :=
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(λ (f g : A → B) (p : f ∼ g),
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let d := λ (x : A), dpair (f x , f x) idp in
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let e := λ (x : A), dpair (f x , g x) (p x) in
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let precomp1 := compose (pr₁ ∘ dpr1) in
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have equiv1 [visible] : IsEquiv precomp1,
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from @isequiv_src_compose A B,
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have equiv2 [visible] : Π x y, IsEquiv (ap precomp1),
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from IsEquiv.ap_closed precomp1,
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have H' : Π (x y : A → diagonal B),
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pr₁ ∘ dpr1 ∘ x ≈ pr₁ ∘ dpr1 ∘ y → x ≈ y,
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from (λ x y, IsEquiv.inv (ap precomp1)),
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have eq2 : pr₁ ∘ dpr1 ∘ d ≈ pr₁ ∘ dpr1 ∘ e,
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from idp,
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have eq0 : d ≈ e,
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from H' d e eq2,
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have eq1 : (pr₂ ∘ dpr1) ∘ d ≈ (pr₂ ∘ dpr1) ∘ e,
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from ap _ eq0,
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eq1
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)
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end
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context
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universe variables l
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parameters {ua2 : ua_type.{l+2}} {ua3 : ua_type.{l+3}}
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-- Now we use this to prove weak funext, which as we know
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-- implies (with dependent eta) also the strong dependent funext.
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theorem ua_implies_weak_funext : weak_funext.{l} :=
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(λ (A : Type.{l+1}) (P : A → Type.{l+2}) allcontr,
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let U := (λ (x : A), unit) in
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have pequiv : Π (x : A), P x ≃ U x,
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from (λ x, @equiv_contr_unit(P x) (allcontr x)),
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have psim : Π (x : A), P x ≈ U x,
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from (λ x, @IsEquiv.inv _ _
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(@equiv_path (P x) (U x)) (ua2 (P x) (U x)) (pequiv x)),
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have p : P ≈ U,
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from @ua_implies_funext_nondep.{l+2 l+1} ua3 A Type.{l+2} P U psim,
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have tU' : is_contr (A → unit),
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from is_contr.mk (λ x, ⋆)
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(λ f, @ua_implies_funext_nondep ua2 A unit (λ x, ⋆) f
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(λ x, unit.rec_on (f x) idp)),
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have tU : is_contr (Π x, U x),
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from tU',
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have tlast : is_contr (Πx, P x),
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from path.transport _ (p⁻¹) tU,
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tlast
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)
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end
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exit
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-- In the following we will proof function extensionality using the univalence axiom
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-- TODO: check out why I have to generalize on A and P here
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definition ua_implies_funext_type {ua : ua_type} : @funext_type :=
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(λ A P, weak_funext_implies_funext (@ua_implies_visible]weak_funext ua))
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