189 lines
6.9 KiB
Text
189 lines
6.9 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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open path function
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-- Equivalences
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-- ------------
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definition 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 [class] {A B : Type} (f : A → B) :=
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IsEquiv_mk : Π
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(inv : B → A)
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(retr : Sect inv f)
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(sect : Sect f inv)
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(adj : Πx, retr (f x) ≈ ap f (sect x)),
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IsEquiv f
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namespace IsEquiv
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definition inv {A B : Type} {f : A → B} (H : IsEquiv f) : B → A :=
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IsEquiv.rec (λinv retr sect adj, inv) H
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-- TODO: note: does not type check without giving the type
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definition retr {A B : Type} {f : A → B} (H : IsEquiv f) : Sect (inv H) f :=
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IsEquiv.rec (λinv retr sect adj, retr) H
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definition sect {A B : Type} {f : A → B} (H : IsEquiv f) : Sect f (inv H) :=
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IsEquiv.rec (λinv retr sect adj, sect) H
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definition adj {A B : Type} {f : A → B} (H : IsEquiv f) :
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Πx, retr H (f x) ≈ ap f (sect H x) :=
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IsEquiv.rec (λinv retr sect adj, adj) H
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end IsEquiv
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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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namespace Equiv
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definition equiv_fun [coercion] {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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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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infix `≃`:25 := Equiv
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notation e `⁻¹` := IsEquiv.inv e
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end Equiv
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-- Some instances and closure properties of equivalences
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namespace IsEquiv
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variables {A B C : Type} {f : A → B} {g : B → C} {f' : A → B}
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-- The identity function is an equivalence.
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definition idIsEquiv [instance] : (@IsEquiv A A id) := IsEquiv_mk id (λa, idp) (λa, idp) (λa, idp)
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-- The composition of two equivalences is, again, an equivalence.
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definition comp_closed [instance] (Hf : IsEquiv f) (Hg : IsEquiv g) : (IsEquiv (g ∘ f)) :=
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IsEquiv_mk ((inv Hf) ∘ (inv Hg))
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(λc, ap g (retr Hf ((inv Hg) c)) @ retr Hg c)
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(λa, ap (inv Hf) (sect Hg (f a)) @ sect Hf a)
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(λa, (whiskerL _ (adj Hg (f a))) @
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(ap_pp g _ _)^ @
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ap02 g (concat_A1p (retr Hf) (sect Hg (f a))^ @
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(ap_compose (inv Hf) f _ @@ adj Hf a) @
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(ap_pp f _ _)^
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) @
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(ap_compose f g _)^
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)
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-- Any function equal to an equivalence is an equivlance as well.
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definition path_closed (Hf : IsEquiv f) (Heq : f ≈ f') : (IsEquiv f') :=
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path.induction_on Heq Hf
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-- Any function pointwise equal to an equivalence is an equivalence as well.
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definition homotopic (Hf : IsEquiv f) (Heq : f ∼ f') : (IsEquiv f') :=
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let sect' := (λ b, (Heq (inv Hf b))^ @ retr Hf b) in
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let retr' := (λ a, (ap (inv Hf) (Heq a))^ @ sect Hf a) in
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let adj' := (λ (a : A),
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let ff'a := Heq a in
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let invf := inv Hf in
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let secta := sect Hf a in
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let retrfa := retr Hf (f a) in
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let retrf'a := retr Hf (f' a) in
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have eq1 : ap f secta @ ff'a ≈ ap f (ap invf ff'a) @ retr Hf (f' a),
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from calc ap f secta @ ff'a
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≈ retrfa @ ff'a : (ap _ (adj Hf _ ))^
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... ≈ ap (f ∘ invf) ff'a @ retrf'a : !concat_A1p^
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... ≈ ap f (ap invf ff'a) @ retr Hf (f' a) : {ap_compose invf f ff'a},
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have eq2 : retrf'a ≈ Heq (invf (f' a)) @ ((ap f' (ap invf ff'a))^ @ ap f' secta),
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from calc retrf'a
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≈ (ap f (ap invf ff'a))^ @ (ap f secta @ ff'a) : moveL_Vp _ _ _ (eq1^)
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... ≈ ap f (ap invf ff'a)^ @ (ap f secta @ Heq a) : {ap_V invf ff'a}
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... ≈ ap f (ap invf ff'a)^ @ (Heq (invf (f a)) @ ap f' secta) : {!concat_Ap}
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... ≈ ap f (ap invf ff'a)^ @ Heq (invf (f a)) @ ap f' secta : {!concat_pp_p^}
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... ≈ ap f ((ap invf ff'a)^) @ Heq (invf (f a)) @ ap f' secta : {!ap_V^}
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... ≈ Heq (invf (f' a)) @ ap f' ((ap invf ff'a)^) @ ap f' secta : {!concat_Ap}
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... ≈ Heq (invf (f' a)) @ (ap f' (ap invf ff'a))^ @ ap f' secta : {!ap_V}
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... ≈ Heq (invf (f' a)) @ ((ap f' (ap invf ff'a))^ @ ap f' secta) : !concat_pp_p,
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have eq3 : (Heq (invf (f' a)))^ @ retr Hf (f' a) ≈ ap f' ((ap invf ff'a)^ @ secta),
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from calc (Heq (invf (f' a)))^ @ retr Hf (f' a)
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≈ (ap f' (ap invf ff'a))^ @ ap f' secta : moveR_Vp _ _ _ eq2
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... ≈ (ap f' ((ap invf ff'a)^)) @ ap f' secta : {!ap_V^}
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... ≈ ap f' ((ap invf ff'a)^ @ secta) : !ap_pp^,
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eq3) in
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IsEquiv_mk (inv Hf) sect' retr' adj'
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--TODO: Maybe wait until rewrite rules are available.
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definition inv_closed (Hf : IsEquiv f) : (IsEquiv (inv Hf)) :=
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IsEquiv_mk sorry sorry sorry sorry
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definition cancel_R (Hf : IsEquiv f) (Hgf : IsEquiv (g ∘ f)) : (IsEquiv g) :=
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homotopic (comp_closed (inv_closed Hf) Hgf) (λb, ap g (retr Hf b))
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definition cancel_L (Hg : IsEquiv g) (Hgf : IsEquiv (g ∘ f)) : (IsEquiv f) :=
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homotopic (comp_closed Hgf (inv_closed Hg)) (λa, sect Hg (f a))
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definition transport (P : A → Type) {x y : A} (p : x ≈ y) : (IsEquiv (transport P p)) :=
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IsEquiv_mk (transport P (p^)) (transport_pV P p) (transport_Vp P p) (transport_pVp P p)
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--Rewrite rules
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section
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variables {Hf : IsEquiv f}
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definition moveR_M {x : A} {y : B} (p : x ≈ (inv Hf) y) : (f x ≈ y) :=
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ap f p @ retr Hf y
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definition moveL_M {x : A} {y : B} (p : (inv Hf) y ≈ x) : (y ≈ f x) :=
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(moveR_M (p^))^
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definition moveR_V {x : B} {y : A} (p : x ≈ f y) : (inv Hf) x ≈ y :=
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ap (inv Hf) p @ sect Hf y
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definition moveL_V {x : B} {y : A} (p : f y ≈ x) : y ≈ (inv Hf) x :=
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(moveR_V (p^))^
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end
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end IsEquiv
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namespace Equiv
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variables {A B C : Type} (eqf : A ≃ B)
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theorem id : A ≃ A := Equiv_mk id IsEquiv.idIsEquiv
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theorem compose (eqg: B ≃ C) : A ≃ C :=
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Equiv_mk ((equiv_fun eqg) ∘ (equiv_fun eqf))
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(IsEquiv.comp_closed (equiv_isequiv eqf) (equiv_isequiv eqg))
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check IsEquiv.path_closed
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theorem path_closed (f' : A → B) (Heq : equiv_fun eqf ≈ f') : A ≃ B :=
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Equiv_mk f' (IsEquiv.path_closed (equiv_isequiv eqf) Heq)
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theorem inv_closed : B ≃ A :=
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Equiv_mk (IsEquiv.inv (equiv_isequiv eqf)) (IsEquiv.inv_closed (equiv_isequiv eqf))
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theorem cancel_L {f : A → B} {g : B → C}
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(Hf : IsEquiv f) (Hgf : IsEquiv (g ∘ f)) : B ≃ C :=
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Equiv_mk g (IsEquiv.cancel_R _ _)
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theorem cancel_R {f : A → B} {g : B → C}
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(Hg : IsEquiv g) (Hgf : IsEquiv (g ∘ f)) : A ≃ B :=
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Equiv_mk f (!IsEquiv.cancel_L _ _)
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theorem transport (P : A → Type) {x y : A} {p : x ≈ y} : (P x) ≃ (P y) :=
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Equiv_mk (transport P p) (IsEquiv.transport P p)
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end Equiv
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