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