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chore(library/hott) fix universe issue. note: this should be fixed when contr is not bound to universe level 1 anymore

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Jakob von Raumer 2014-11-14 01:03:17 -05:00 committed by Leonardo de Moura
parent 992aad9661
commit 59fbe8b53e
1 changed files with 46 additions and 32 deletions
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78
library/hott/funext_from_ua.lean
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@ -23,37 +23,50 @@ context
-- TODO base this theorem on UA instead of FunExt.
-- IsEquiv.postcompose relies on FunExt!
protected theorem ua_isequiv_postcompose {A B C : Type.{1}} {w : A → B} {H0 : IsEquiv w}
: IsEquiv (@compose C A B w)
:= IsEquiv.adjointify (@compose C A B w)
(@compose C B A (IsEquiv.inv w))
(λ (x : C → B),
let w' := Equiv.mk w H0 in
have foo : Equiv.equiv_fun w' ≈ w,
from idp,
have eqretr : equiv_path (equiv_path⁻¹ w') ≈ w',
from (@retr _ _ (@equiv_path A B) (ua A B) w'),
have eqinv : A ≈ B,
from (@inv _ _ (@equiv_path A B) (ua A B) w'),
have thoseeqs [visible] : Π (p : A ≈ B), IsEquiv (Equiv.equiv_fun (equiv_path p)),
from (λp, Equiv.equiv_isequiv (equiv_path p)),
have eqp : Π (p : A ≈ B) (x : C → B), equiv_path p ∘ ((equiv_path p)⁻¹ ∘ x) ≈ x,
from (λ p,
(@path.rec_on Type.{1} A
(λ B' p', Π (x' : C → B'), (@equiv_path A B' p') ∘ ((equiv_path p')⁻¹ ∘ x') ≈ x')
B p (λ x', idp))
),
--have eqfin : equiv_path eqinv ∘ ((equiv_path eqinv)⁻¹ eqinv ∘ x) ≈ x,
-- from eqp eqinv,
sorry
)
(λ x, sorry)
exit
: IsEquiv (@compose C A B w) :=
let w' := Equiv.mk w H0 in
let eqinv : A ≈ B := (equiv_path⁻¹ w') in
let eq' := equiv_path eqinv in
IsEquiv.adjointify (@compose C A B w)
(@compose C B A (IsEquiv.inv w))
(λ (x : C → B),
have eqretr : eq' ≈ w',
from (@retr _ _ (@equiv_path A B) (ua A B) w'),
have invs_eq : (equiv_fun eq')⁻¹ ≈ (equiv_fun w')⁻¹,
from inv_eq eq' w' eqretr,
have eqfin : (equiv_fun eq') ∘ ((equiv_fun eq')⁻¹ ∘ x) ≈ x,
from (λ p,
(@path.rec_on Type.{1} A
(λ B' p', Π (x' : C → B'), (equiv_fun (equiv_path p'))
∘ ((equiv_fun (equiv_path p'))⁻¹ ∘ x') ≈ x')
B p (λ x', idp))
) eqinv x,
have eqfin' : (equiv_fun w') ∘ ((equiv_fun eq')⁻¹ ∘ x) ≈ x,
from eqretr ▹ eqfin,
have eqfin'' : (equiv_fun w') ∘ ((equiv_fun w')⁻¹ ∘ x) ≈ x,
from invs_eq ▹ eqfin',
eqfin''
)
(λ (x : C → A),
have eqretr : eq' ≈ w',
from (@retr _ _ (@equiv_path A B) (ua A B) w'),
have invs_eq : (equiv_fun eq')⁻¹ ≈ (equiv_fun w')⁻¹,
from inv_eq eq' w' eqretr,
have eqfin : (equiv_fun eq')⁻¹ ∘ ((equiv_fun eq') ∘ x) ≈ x,
from (λ p, path.rec_on p idp) eqinv,
have eqfin' : (equiv_fun eq')⁻¹ ∘ ((equiv_fun w') ∘ x) ≈ x,
from eqretr ▹ eqfin,
have eqfin'' : (equiv_fun w')⁻¹ ∘ ((equiv_fun w') ∘ x) ≈ x,
from invs_eq ▹ eqfin',
eqfin''
)
-- We are ready to prove functional extensionality,
-- starting with the naive non-dependent version.
protected definition diagonal [reducible] (B : Type) : Type
:= Σ xy : B × B, pr₁ xy ≈ pr₂ xy
protected definition isequiv_src_compose {A B C : Type}
protected definition isequiv_src_compose {A B C : Type.{1}}
: @IsEquiv (A → diagonal B)
(A → B)
(compose (pr₁ ∘ dpr1))
@ -64,7 +77,7 @@ exit
(λ xy, prod.rec_on xy
(λ b c p, path.rec_on p idp))))
protected definition isequiv_tgt_compose {A B C : Type}
protected definition isequiv_tgt_compose {A B C : Type.{1}}
: @IsEquiv (A → diagonal B)
(A → B)
(compose (pr₂ ∘ dpr1))
@ -75,7 +88,7 @@ exit
(λ xy, prod.rec_on xy
(λ b c p, path.rec_on p idp))))
theorem ua_implies_funext_nondep {A B : Type}
theorem ua_implies_funext_nondep {A B : Type.{1}}
: Π {f g : A → B}, f ∼ g → f ≈ g
:= (λ (f g : A → B) (p : f ∼ g),
let d := λ (x : A), dpair (f x , f x) idp in
@ -101,12 +114,13 @@ end
context
universe l
parameters {ua1 ua2 : ua_type.{1}}
parameters {ua1 : ua_type.{1}} {ua2 : ua_type.{2}}
-- Now we use this to prove weak funext, which as we know
-- implies (with dependent eta) also the strong dependent funext.
set_option pp.universes true
theorem ua_implies_weak_funext : weak_funext
:= (λ A P allcontr,
:= (λ (A : Type.{1}) (P : A → Type.{1}) allcontr,
let U := (λ (x : A), unit) in
have pequiv : Πx, P x ≃ U x,
from (λ x, @equiv_contr_unit (P x) (allcontr x)),
@ -114,10 +128,10 @@ context
from (λ x, @IsEquiv.inv _ _
(@equiv_path.{1} (P x) (U x)) (ua1 (P x) (U x)) (pequiv x)),
have p : P ≈ U,
from ua_implies_funext_nondep psim,
from sorry, --ua_implies_funext_nondep psim,
have tU' : is_contr (A → unit),
from is_contr.mk (λ x, ⋆)
(λ f, ua_implies_funext_nondep
(λ f, @ua_implies_funext_nondep ua1 _ _ _ _
(λ x, unit.rec_on (f x) idp)),
have tU : is_contr (Πx, U x),
from tU',