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chore(library/hott) move theorem about precomposition to its own file

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Jakob von Raumer 2014-11-04 14:23:37 -05:00 committed by Leonardo de Moura
parent 0ed046ed80
commit efa33c5b52
2 changed files with 46 additions and 35 deletions
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35
library/hott/equiv.lean
View file

@ -212,41 +212,6 @@ namespace IsEquiv
end end
--If pre- or post-composing with a function is always an equivalence,
--then that function is also an equivalence. It's convenient to know
--that we only need to assume the equivalence when the other type is
--the domain or the codomain.
section
definition precomp (C : Type) (h : B → C) : A → C := h ∘ f
definition inv_precomp (C D : Type) (Ceq : IsEquiv (precomp C))
(Deq : IsEquiv (@precomp A B f D)) (k : C → D) (h : A → C) :
k ∘ (inv (precomp C)) h ≈ (inv (precomp D)) (k ∘ h) :=
let invD := inv (precomp D) in
let invC := inv (precomp C) in
have eq1 : invD (k ∘ h) ≈ k ∘ (invC h),
from calc invD (k ∘ h) ≈ invD (k ∘ (precomp C (invC h))) : retr (precomp C) h
... ≈ k ∘ (invC h) : !sect,
eq1⁻¹
definition isequiv_precompose (Aeq : IsEquiv (@precomp A B f A))
(Beq : IsEquiv (@precomp A B f B)) : (IsEquiv f) :=
let invA := inv (precomp A) in
let invB := inv (precomp B) in
let sect' : Sect (invA id) f := (λx,
calc f (invA id x) ≈ (f ∘ invA id) x : idp
... ≈ invB (f ∘ id) x : apD10 (!inv_precomp)
... ≈ invB (@precomp A B f B id) x : idp
... ≈ x : apD10 (sect (precomp B) id))
in
let retr' : Sect f (invA id) := (λx,
calc invA id (f x) ≈ @precomp A B f A (invA id) x : idp
... ≈ x : apD10 (retr (precomp A) id)) in
adjointify f (invA id) sect' retr'
end
end IsEquiv end IsEquiv
namespace Equiv namespace Equiv

46
library/hott/equiv_precomp.lean Normal file
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@ -0,0 +1,46 @@
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Jeremy Avigad, Jakob von Raumer
-- Ported from Coq HoTT
import .equiv .funext
open path function
namespace IsEquiv
--If pre- or post-composing with a function is always an equivalence,
--then that function is also an equivalence. It's convenient to know
--that we only need to assume the equivalence when the other type is
--the domain or the codomain.
context
parameters {A B : Type} (f : A → B)
definition precomp (C : Type) (h : B → C) : A → C := h ∘ f
definition inv_precomp (C D : Type) (Ceq : IsEquiv (precomp C))
(Deq : IsEquiv (precomp D)) (k : C → D) (h : A → C) :
k ∘ (inv (precomp C)) h ≈ (inv (precomp D)) (k ∘ h) :=
let invD := inv (precomp D) in
let invC := inv (precomp C) in
have eq1 : invD (k ∘ h) ≈ k ∘ (invC h),
from calc invD (k ∘ h) ≈ invD (k ∘ (precomp C (invC h))) : retr (precomp C) h
... ≈ k ∘ (invC h) : !sect,
eq1⁻¹
definition isequiv_precompose (Aeq : IsEquiv (precomp A))
(Beq : IsEquiv (precomp B)) : (IsEquiv f) :=
let invA := inv (precomp A) in
let invB := inv (precomp B) in
let sect' : Sect (invA id) f := (λx,
calc f (invA id x) ≈ (f ∘ invA id) x : idp
... ≈ invB (f ∘ id) x : apD10 (!inv_precomp)
... ≈ invB (precomp B id) x : idp
... ≈ x : apD10 (sect (precomp B) id))
in
let retr' : Sect f (invA id) := (λx,
calc invA id (f x) ≈ precomp A (invA id) x : idp
... ≈ x : apD10 (retr (precomp A) id)) in
adjointify f (invA id) sect' retr'
end
end IsEquiv