61 lines
1.9 KiB
Text
61 lines
1.9 KiB
Text
-- 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
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-- Ported from Coq HoTT
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import .path
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-- Equivalences
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-- ------------
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abbreviation Sect {A B : Type} (s : A → B) (r : B → A) := Πx : A, r (s x) ≈ x
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-- -- TODO: need better means of declaring structures
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-- -- TODO: note that Coq allows projections to be declared to be coercions on the fly
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-- Structure IsEquiv
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inductive IsEquiv {A B : Type} (f : A → B) :=
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IsEquiv_mk : Π
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(equiv_inv : B → A)
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(eisretr : Sect equiv_inv f)
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(eissect : Sect f equiv_inv)
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(eisadj : Πx, eisretr (f x) ≈ ap f (eissect x)),
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IsEquiv f
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definition equiv_inv {A B : Type} {f : A → B} (H : IsEquiv f) : B → A :=
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IsEquiv_rec (λequiv_inv eisretr eissect eisadj, equiv_inv) H
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-- TODO: note: does not type check without giving the type
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definition eisretr {A B : Type} {f : A → B} (H : IsEquiv f) : Sect (equiv_inv H) f :=
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IsEquiv_rec (λequiv_inv eisretr eissect eisadj, eisretr) H
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definition eissect {A B : Type} {f : A → B} (H : IsEquiv f) : Sect f (equiv_inv H) :=
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IsEquiv_rec (λequiv_inv eisretr eissect eisadj, eissect) H
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definition eisadj {A B : Type} {f : A → B} (H : IsEquiv f) :
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Πx, eisretr H (f x) ≈ ap f (eissect H x) :=
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IsEquiv_rec (λequiv_inv eisretr eissect eisadj, eisadj) H
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-- Structure Equiv
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inductive Equiv (A B : Type) : Type :=
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Equiv_mk : Π
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(equiv_fun : A → B)
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(equiv_isequiv : IsEquiv equiv_fun),
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Equiv A B
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definition equiv_fun {A B : Type} (e : Equiv A B) : A → B :=
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Equiv_rec (λequiv_fun equiv_isequiv, equiv_fun) e
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-- TODO: use a type class instead?
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coercion equiv_fun : Equiv
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definition equiv_isequiv [coercion] {A B : Type} (e : Equiv A B) : IsEquiv (equiv_fun e) :=
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Equiv_rec (λequiv_fun equiv_isequiv, equiv_isequiv) e
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-- coercion equiv_isequiv
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-- TODO: better symbol
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infix `<~>`:25 := Equiv
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notation e `⁻¹`:75 := equiv_inv e
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