299 lines
12 KiB
Text
299 lines
12 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, Jakob von Raumer
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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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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 f) 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 f) :=
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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 f (f x) ≈ ap f (sect f x) :=
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IsEquiv.rec (λinv retr sect adj, adj) H
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postfix `⁻¹` := inv
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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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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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--Note: No coercion here
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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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definition equiv_isequiv [instance] {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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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 id_closed [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 f) ∘ (inv g))
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(λc, ap g (retr f (g⁻¹ c)) ⬝ retr g c)
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(λa, ap (inv f) (sect g (f a)) ⬝ sect f a)
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(λa, (whiskerL _ (adj g (f a))) ⬝
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(ap_pp g _ _)⁻¹ ⬝
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ap02 g (concat_A1p (retr f) (sect g (f a))⁻¹ ⬝
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(ap_compose (inv f) f _ ◾ adj f a) ⬝
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(ap_pp f _ _)⁻¹
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) ⬝
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(ap_compose f g _)⁻¹
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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.rec_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 f b))⁻¹ ⬝ retr f b) in
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let retr' := (λ a, (ap (inv f) (Heq a))⁻¹ ⬝ sect f 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 f in
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let secta := sect f a in
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let retrfa := retr f (f a) in
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let retrf'a := retr f (f' a) in
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have eq1 : _ ≈ _,
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from calc ap f secta ⬝ ff'a
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≈ retrfa ⬝ ff'a : (ap _ (adj f _ ))⁻¹
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... ≈ ap (f ∘ invf) ff'a ⬝ retrf'a : !concat_A1p⁻¹
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... ≈ ap f (ap invf ff'a) ⬝ retr f (f' a) : {ap_compose invf f _},
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have eq2 : _ ≈ _,
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from calc retrf'a
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≈ (ap f (ap invf ff'a))⁻¹ ⬝ (ap f secta ⬝ ff'a) : moveL_Vp _ _ _ (eq1⁻¹)
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... ≈ ap f (ap invf ff'a)⁻¹ ⬝ (ap f secta ⬝ Heq a) : {ap_V invf ff'a}
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... ≈ ap f (ap invf ff'a)⁻¹ ⬝ (Heq (invf (f a)) ⬝ ap f' secta) : {!concat_Ap}
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... ≈ (ap f (ap invf ff'a)⁻¹ ⬝ Heq (invf (f a))) ⬝ ap f' secta : {!concat_pp_p⁻¹}
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... ≈ (ap f ((ap invf ff'a)⁻¹) ⬝ Heq (invf (f a))) ⬝ ap f' secta : {!ap_V⁻¹}
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... ≈ (Heq (invf (f' a)) ⬝ ap f' ((ap invf ff'a)⁻¹)) ⬝ ap f' secta : {!concat_Ap}
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... ≈ (Heq (invf (f' a)) ⬝ (ap f' (ap invf ff'a))⁻¹) ⬝ ap f' secta : {!ap_V}
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... ≈ Heq (invf (f' a)) ⬝ ((ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta) : !concat_pp_p,
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have eq3 : _ ≈ _,
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from calc (Heq (invf (f' a)))⁻¹ ⬝ retr f (f' a)
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≈ (ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta : moveR_Vp _ _ _ eq2
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... ≈ (ap f' ((ap invf ff'a)⁻¹)) ⬝ ap f' secta : {!ap_V⁻¹}
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... ≈ ap f' ((ap invf ff'a)⁻¹ ⬝ secta) : !ap_pp⁻¹,
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eq3) in
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IsEquiv.mk (inv f) sect' retr' adj'
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end IsEquiv
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namespace IsEquiv
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variables {A B : Type} (f : A → B) (g : B → A)
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(ret : Sect g f) (sec : Sect f g)
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context
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set_option unifier.max_steps 30000
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--To construct an equivalence it suffices to state the proof that the inverse is a quasi-inverse.
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definition adjointify : IsEquiv f :=
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let sect' := (λx, ap g (ap f (inverse (sec x))) ⬝ ap g (ret (f x)) ⬝ sec x) in
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let adj' := (λ (a : A),
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let fgretrfa := ap f (ap g (ret (f a))) in
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let fgfinvsect := ap f (ap g (ap f ((sec a)⁻¹))) in
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let fgfa := f (g (f a)) in
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let retrfa := ret (f a) in
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have eq1 : ap f (sec a) ≈ _,
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from calc ap f (sec a)
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≈ idp ⬝ ap f (sec a) : !concat_1p⁻¹
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... ≈ (ret (f a) ⬝ (ret (f a)⁻¹)) ⬝ ap f (sec a) : {!concat_pV⁻¹}
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... ≈ ((ret (fgfa))⁻¹ ⬝ ap (f ∘ g) (ret (f a))) ⬝ ap f (sec a) : {!concat_pA1⁻¹}
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... ≈ ((ret (fgfa))⁻¹ ⬝ fgretrfa) ⬝ ap f (sec a) : {ap_compose g f _}
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... ≈ (ret (fgfa))⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a)) : !concat_pp_p,
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have eq2 : ap f (sec a) ⬝ idp ≈ (ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a)),
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from !concat_p1 ⬝ eq1,
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have eq3 : idp ≈ _,
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from calc idp
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≈ (ap f (sec a))⁻¹ ⬝ ((ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a))) : moveL_Vp _ _ _ eq2
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... ≈ (ap f (sec a)⁻¹ ⬝ (ret fgfa)⁻¹) ⬝ (fgretrfa ⬝ ap f (sec a)) : !concat_p_pp
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... ≈ (ap f ((sec a)⁻¹) ⬝ (ret fgfa)⁻¹) ⬝ (fgretrfa ⬝ ap f (sec a)) : {!ap_V⁻¹}
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... ≈ ((ap f ((sec a)⁻¹) ⬝ (ret fgfa)⁻¹) ⬝ fgretrfa) ⬝ ap f (sec a) : !concat_p_pp
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... ≈ ((retrfa⁻¹ ⬝ ap (f ∘ g) (ap f ((sec a)⁻¹))) ⬝ fgretrfa) ⬝ ap f (sec a) : {!concat_pA1⁻¹}
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... ≈ ((retrfa⁻¹ ⬝ fgfinvsect) ⬝ fgretrfa) ⬝ ap f (sec a) : {ap_compose g f _}
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... ≈ (retrfa⁻¹ ⬝ (fgfinvsect ⬝ fgretrfa)) ⬝ ap f (sec a) : {!concat_p_pp⁻¹}
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... ≈ retrfa⁻¹ ⬝ ap f (ap g (ap f ((sec a)⁻¹)) ⬝ ap g (ret (f a))) ⬝ ap f (sec a) : {!ap_pp⁻¹}
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... ≈ retrfa⁻¹ ⬝ (ap f (ap g (ap f ((sec a)⁻¹)) ⬝ ap g (ret (f a))) ⬝ ap f (sec a)) : !concat_p_pp⁻¹
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... ≈ retrfa⁻¹ ⬝ ap f ((ap g (ap f ((sec a)⁻¹)) ⬝ ap g (ret (f a))) ⬝ sec a) : {!ap_pp⁻¹},
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have eq4 : ret (f a) ≈ ap f ((ap g (ap f ((sec a)⁻¹)) ⬝ ap g (ret (f a))) ⬝ sec a),
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from moveR_M1 _ _ eq3,
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eq4) in
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IsEquiv.mk g ret sect' adj'
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end
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end IsEquiv
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namespace IsEquiv
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variables {A B: Type} (f : A → B)
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--The inverse of an equivalence is, again, an equivalence.
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definition inv_closed [instance] [Hf : IsEquiv f] : (IsEquiv (inv f)) :=
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adjointify (inv f) f (sect f) (retr f)
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end IsEquiv
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namespace IsEquiv
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variables {A : Type}
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section
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variables {B C : Type} (f : A → B) {f' : A → B} [Hf : IsEquiv f]
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include Hf
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definition cancel_R (g : B → C) [Hgf : IsEquiv (g ∘ f)] : (IsEquiv g) :=
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homotopic (comp_closed !inv_closed Hgf) (λb, ap g (retr f b))
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definition cancel_L (g : C → A) [Hgf : IsEquiv (f ∘ g)] : (IsEquiv g) :=
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homotopic (comp_closed Hgf !inv_closed) (λa, sect f (g a))
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--Rewrite rules
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definition moveR_M {x : A} {y : B} (p : x ≈ (inv f) y) : (f x ≈ y) :=
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(ap f p) ⬝ (retr f y)
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definition moveL_M {x : A} {y : B} (p : (inv f) y ≈ x) : (y ≈ f x) :=
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(moveR_M f (p⁻¹))⁻¹
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definition moveR_V {x : B} {y : A} (p : x ≈ f y) : (inv f) x ≈ y :=
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ap (inv f) p ⬝ sect f y
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definition moveL_V {x : B} {y : A} (p : f y ≈ x) : y ≈ (inv f) x :=
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(moveR_V f (p⁻¹))⁻¹
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definition ap_closed [instance] (x y : A) : IsEquiv (ap f) :=
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adjointify (ap f)
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(λq, (inverse (sect f x)) ⬝ ap (f⁻¹) q ⬝ sect f y)
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(λq, !ap_pp
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⬝ whiskerR !ap_pp _
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⬝ ((!ap_V ⬝ inverse2 ((adj f _)⁻¹))
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◾ (inverse (ap_compose (f⁻¹) f _))
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◾ (adj f _)⁻¹)
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⬝ concat_pA1_p (retr f) _ _
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⬝ whiskerR !concat_Vp _
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⬝ !concat_1p)
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(λp, whiskerR (whiskerL _ ((ap_compose f (f⁻¹) _)⁻¹)) _
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⬝ concat_pA1_p (sect f) _ _
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⬝ whiskerR !concat_Vp _
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⬝ !concat_1p)
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-- The function equiv_rect says that given an equivalence f : A → B,
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-- and a hypothesis from B, one may always assume that the hypothesis
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-- is in the image of e.
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-- In fibrational terms, if we have a fibration over B which has a section
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-- once pulled back along an equivalence f : A → B, then it has a section
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-- over all of B.
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definition equiv_rect (P : B -> Type) :
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(Πx, P (f x)) → (Πy, P y) :=
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(λg y, path.transport _ (retr f y) (g (f⁻¹ y)))
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definition equiv_rect_comp (P : B → Type)
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(df : Π (x : A), P (f x)) (x : A) : equiv_rect f P df (f x) ≈ df x :=
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let eq1 := (apD df (sect f x)) in
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calc equiv_rect f P df (f x)
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≈ transport P (retr f (f x)) (df (f⁻¹ (f x))) : idp
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... ≈ transport P (ap f (sect f x)) (df (f⁻¹ (f x))) : adj f
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... ≈ transport (P ∘ f) (sect f x) (df (f⁻¹ (f x))) : transport_compose
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... ≈ df x : eq1
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end
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--Transporting is an equivalence
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protected definition transport [instance] (P : A → Type) {x y : A} (p : x ≈ y) : (IsEquiv (transport P p)) :=
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IsEquiv.mk (transport P (p⁻¹)) (transport_pV P p) (transport_Vp P p) (transport_pVp P p)
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end IsEquiv
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namespace Equiv
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context
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parameters {A B C : Type} (eqf : A ≃ B)
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private definition f : A → B := equiv_fun eqf
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private definition Hf : IsEquiv f := equiv_isequiv eqf
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protected definition id : A ≃ A := Equiv.mk id IsEquiv.id_closed
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protected theorem compose (eqg: B ≃ C) : A ≃ C :=
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Equiv.mk ((equiv_fun eqg) ∘ f)
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(IsEquiv.comp_closed Hf (equiv_isequiv eqg))
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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 Hf Heq)
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theorem inv_closed : B ≃ A :=
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Equiv.mk (IsEquiv.inv f) !IsEquiv.inv_closed
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theorem cancel_R {g : B → C} (Hgf : IsEquiv (g ∘ f)) : B ≃ C :=
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Equiv.mk g (IsEquiv.cancel_R f _)
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theorem cancel_L {g : C → A} (Hgf : IsEquiv (f ∘ g)) : C ≃ A :=
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Equiv.mk g (IsEquiv.cancel_L f _)
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protected 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
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context
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parameters {A B : Type} (eqf eqg : A ≃ B)
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private definition Hf [instance] : IsEquiv (equiv_fun eqf) := equiv_isequiv eqf
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private definition Hg [instance] : IsEquiv (equiv_fun eqg) := equiv_isequiv eqg
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theorem inv_eq (p : eqf ≈ eqg)
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: IsEquiv.inv (equiv_fun eqf) ≈ IsEquiv.inv (equiv_fun eqg) :=
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path.rec_on p idp
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end
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-- calc enviroment
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-- Note: Calculating with substitutions needs univalence
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calc_trans compose
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calc_refl id
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calc_symm inv_closed
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end Equiv
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