lean2/library/hott/equiv.lean
Floris van Doorn 107a9cf8e4 feat(library): port more of truncation library from Coq HoTT
Everything directly about truncations in the basic truncation library is ported.
Some theorems about other structures still need to be ported.
Also made some minor changes in hott.equiv
2014-11-08 19:12:54 -08:00

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-- 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 .path
open path function
-- Equivalences
-- ------------
definition Sect {A B : Type} (s : A → B) (r : B → A) := Πx : A, r (s x) ≈ x
-- -- TODO: need better means of declaring structures
-- -- TODO: note that Coq allows projections to be declared to be coercions on the fly
-- Structure IsEquiv
inductive IsEquiv [class] {A B : Type} (f : A → B) :=
mk : Π
(inv : B → A)
(retr : Sect inv f)
(sect : Sect f inv)
(adj : Πx, retr (f x) ≈ ap f (sect x)),
IsEquiv f
namespace IsEquiv
definition inv {A B : Type} (f : A → B) [H : IsEquiv f] : B → A :=
IsEquiv.rec (λinv retr sect adj, inv) H
-- TODO: note: does not type check without giving the type
definition retr {A B : Type} (f : A → B) [H : IsEquiv f] : Sect (inv f) f :=
IsEquiv.rec (λinv retr sect adj, retr) H
definition sect {A B : Type} (f : A → B) [H : IsEquiv f] : Sect f (inv f) :=
IsEquiv.rec (λinv retr sect adj, sect) H
definition adj {A B : Type} (f : A → B) [H : IsEquiv f] :
Πx, retr f (f x) ≈ ap f (sect f x) :=
IsEquiv.rec (λinv retr sect adj, adj) H
postfix `⁻¹` := inv
end IsEquiv
-- Structure Equiv
inductive Equiv (A B : Type) : Type :=
mk : Π
(equiv_fun : A → B)
(equiv_isequiv : IsEquiv equiv_fun),
Equiv A B
namespace Equiv
definition equiv_fun [coercion] {A B : Type} (e : Equiv A B) : A → B :=
Equiv.rec (λequiv_fun equiv_isequiv, equiv_fun) e
definition equiv_isequiv [instance] {A B : Type} (e : Equiv A B) : IsEquiv (equiv_fun e) :=
Equiv.rec (λequiv_fun equiv_isequiv, equiv_isequiv) e
infix `≃`:25 := Equiv
end Equiv
-- Some instances and closure properties of equivalences
namespace IsEquiv
variables {A B C : Type} {f : A → B} {g : B → C} {f' : A → B}
-- The identity function is an equivalence.
definition id_closed [instance] : (@IsEquiv A A id) := IsEquiv.mk id (λa, idp) (λa, idp) (λa, idp)
-- The composition of two equivalences is, again, an equivalence.
definition comp_closed [instance] (Hf : IsEquiv f) (Hg : IsEquiv g) : (IsEquiv (g ∘ f)) :=
IsEquiv.mk ((inv f) ∘ (inv g))
(λc, ap g (retr f (g⁻¹ c)) ⬝ retr g c)
(λa, ap (inv f) (sect g (f a)) ⬝ sect f a)
(λa, (whiskerL _ (adj g (f a))) ⬝
(ap_pp g _ _)⁻¹ ⬝
ap02 g (concat_A1p (retr f) (sect g (f a))⁻¹ ⬝
(ap_compose (inv f) f _ ◾ adj f a) ⬝
(ap_pp f _ _)⁻¹
) ⬝
(ap_compose f g _)⁻¹
)
-- Any function equal to an equivalence is an equivlance as well.
definition path_closed (Hf : IsEquiv f) (Heq : f ≈ f') : (IsEquiv f') :=
path.rec_on Heq Hf
-- Any function pointwise equal to an equivalence is an equivalence as well.
definition homotopic (Hf : IsEquiv f) (Heq : f f') : (IsEquiv f') :=
let sect' := (λ b, (Heq (inv f b))⁻¹ ⬝ retr f b) in
let retr' := (λ a, (ap (inv f) (Heq a))⁻¹ ⬝ sect f a) in
let adj' := (λ (a : A),
let ff'a := Heq a in
let invf := inv f in
let secta := sect f a in
let retrfa := retr f (f a) in
let retrf'a := retr f (f' a) in
have eq1 : _ ≈ _,
from calc ap f secta ⬝ ff'a
≈ retrfa ⬝ ff'a : (ap _ (adj f _ ))⁻¹
... ≈ ap (f ∘ invf) ff'a ⬝ retrf'a : !concat_A1p⁻¹
... ≈ ap f (ap invf ff'a) ⬝ retr f (f' a) : {ap_compose invf f _},
have eq2 : _ ≈ _,
from calc retrf'a
≈ (ap f (ap invf ff'a))⁻¹ ⬝ (ap f secta ⬝ ff'a) : moveL_Vp _ _ _ (eq1⁻¹)
... ≈ ap f (ap invf ff'a)⁻¹ ⬝ (ap f secta ⬝ Heq a) : {ap_V invf ff'a}
... ≈ ap f (ap invf ff'a)⁻¹ ⬝ (Heq (invf (f a)) ⬝ ap f' secta) : {!concat_Ap}
... ≈ (ap f (ap invf ff'a)⁻¹ ⬝ Heq (invf (f a))) ⬝ ap f' secta : {!concat_pp_p⁻¹}
... ≈ (ap f ((ap invf ff'a)⁻¹) ⬝ Heq (invf (f a))) ⬝ ap f' secta : {!ap_V⁻¹}
... ≈ (Heq (invf (f' a)) ⬝ ap f' ((ap invf ff'a)⁻¹)) ⬝ ap f' secta : {!concat_Ap}
... ≈ (Heq (invf (f' a)) ⬝ (ap f' (ap invf ff'a))⁻¹) ⬝ ap f' secta : {!ap_V}
... ≈ Heq (invf (f' a)) ⬝ ((ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta) : !concat_pp_p,
have eq3 : _ ≈ _,
from calc (Heq (invf (f' a)))⁻¹ ⬝ retr f (f' a)
≈ (ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta : moveR_Vp _ _ _ eq2
... ≈ (ap f' ((ap invf ff'a)⁻¹)) ⬝ ap f' secta : {!ap_V⁻¹}
... ≈ ap f' ((ap invf ff'a)⁻¹ ⬝ secta) : !ap_pp⁻¹,
eq3) in
IsEquiv.mk (inv f) sect' retr' adj'
end IsEquiv
namespace IsEquiv
variables {A B : Type} (f : A → B) (g : B → A)
(ret : Sect g f) (sec : Sect f g)
context
set_option unifier.max_steps 30000
--To construct an equivalence it suffices to state the proof that the inverse is a quasi-inverse.
definition adjointify : IsEquiv f :=
let sect' := (λx, ap g (ap f (inverse (sec x))) ⬝ ap g (ret (f x)) ⬝ sec x) in
let adj' := (λ (a : A),
let fgretrfa := ap f (ap g (ret (f a))) in
let fgfinvsect := ap f (ap g (ap f ((sec a)⁻¹))) in
let fgfa := f (g (f a)) in
let retrfa := ret (f a) in
have eq1 : ap f (sec a) ≈ _,
from calc ap f (sec a)
≈ idp ⬝ ap f (sec a) : !concat_1p⁻¹
... ≈ (ret (f a) ⬝ (ret (f a)⁻¹)) ⬝ ap f (sec a) : {!concat_pV⁻¹}
... ≈ ((ret (fgfa))⁻¹ ⬝ ap (f ∘ g) (ret (f a))) ⬝ ap f (sec a) : {!concat_pA1⁻¹}
... ≈ ((ret (fgfa))⁻¹ ⬝ fgretrfa) ⬝ ap f (sec a) : {ap_compose g f _}
... ≈ (ret (fgfa))⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a)) : !concat_pp_p,
have eq2 : ap f (sec a) ⬝ idp ≈ (ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a)),
from !concat_p1 ⬝ eq1,
have eq3 : idp ≈ _,
from calc idp
≈ (ap f (sec a))⁻¹ ⬝ ((ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a))) : moveL_Vp _ _ _ eq2
... ≈ (ap f (sec a)⁻¹ ⬝ (ret fgfa)⁻¹) ⬝ (fgretrfa ⬝ ap f (sec a)) : !concat_p_pp
... ≈ (ap f ((sec a)⁻¹) ⬝ (ret fgfa)⁻¹) ⬝ (fgretrfa ⬝ ap f (sec a)) : {!ap_V⁻¹}
... ≈ ((ap f ((sec a)⁻¹) ⬝ (ret fgfa)⁻¹) ⬝ fgretrfa) ⬝ ap f (sec a) : !concat_p_pp
... ≈ ((retrfa⁻¹ ⬝ ap (f ∘ g) (ap f ((sec a)⁻¹))) ⬝ fgretrfa) ⬝ ap f (sec a) : {!concat_pA1⁻¹}
... ≈ ((retrfa⁻¹ ⬝ fgfinvsect) ⬝ fgretrfa) ⬝ ap f (sec a) : {ap_compose g f _}
... ≈ (retrfa⁻¹ ⬝ (fgfinvsect ⬝ fgretrfa)) ⬝ ap f (sec a) : {!concat_p_pp⁻¹}
... ≈ retrfa⁻¹ ⬝ ap f (ap g (ap f ((sec a)⁻¹)) ⬝ ap g (ret (f a))) ⬝ ap f (sec a) : {!ap_pp⁻¹}
... ≈ retrfa⁻¹ ⬝ (ap f (ap g (ap f ((sec a)⁻¹)) ⬝ ap g (ret (f a))) ⬝ ap f (sec a)) : !concat_p_pp⁻¹
... ≈ retrfa⁻¹ ⬝ ap f ((ap g (ap f ((sec a)⁻¹)) ⬝ ap g (ret (f a))) ⬝ sec a) : {!ap_pp⁻¹},
have eq4 : ret (f a) ≈ ap f ((ap g (ap f ((sec a)⁻¹)) ⬝ ap g (ret (f a))) ⬝ sec a),
from moveR_M1 _ _ eq3,
eq4) in
IsEquiv.mk g ret sect' adj'
end
end IsEquiv
namespace IsEquiv
variables {A B: Type} (f : A → B)
--The inverse of an equivalence is, again, an equivalence.
definition inv_closed [instance] [Hf : IsEquiv f] : (IsEquiv (inv f)) :=
adjointify (inv f) f (sect f) (retr f)
end IsEquiv
namespace IsEquiv
variables {A : Type}
section
variables {B C : Type} (f : A → B) {f' : A → B} [Hf : IsEquiv f]
include Hf
definition cancel_R (g : B → C) [Hgf : IsEquiv (g ∘ f)] : (IsEquiv g) :=
homotopic (comp_closed !inv_closed Hgf) (λb, ap g (retr f b))
definition cancel_L (g : C → A) [Hgf : IsEquiv (f ∘ g)] : (IsEquiv g) :=
homotopic (comp_closed Hgf !inv_closed) (λa, sect f (g a))
--Rewrite rules
definition moveR_M {x : A} {y : B} (p : x ≈ (inv f) y) : (f x ≈ y) :=
(ap f p) ⬝ (retr f y)
definition moveL_M {x : A} {y : B} (p : (inv f) y ≈ x) : (y ≈ f x) :=
(moveR_M f (p⁻¹))⁻¹
definition moveR_V {x : B} {y : A} (p : x ≈ f y) : (inv f) x ≈ y :=
ap (inv f) p ⬝ sect f y
definition moveL_V {x : B} {y : A} (p : f y ≈ x) : y ≈ (inv f) x :=
(moveR_V f (p⁻¹))⁻¹
definition ap_closed [instance] (x y : A) : IsEquiv (ap f) :=
adjointify (ap f)
(λq, (inverse (sect f x)) ⬝ ap (f⁻¹) q ⬝ sect f y)
(λq, !ap_pp
⬝ whiskerR !ap_pp _
⬝ ((!ap_V ⬝ inverse2 ((adj f _)⁻¹))
◾ (inverse (ap_compose (f⁻¹) f _))
◾ (adj f _)⁻¹)
⬝ concat_pA1_p (retr f) _ _
⬝ whiskerR !concat_Vp _
⬝ !concat_1p)
(λp, whiskerR (whiskerL _ ((ap_compose f (f⁻¹) _)⁻¹)) _
⬝ concat_pA1_p (sect f) _ _
⬝ whiskerR !concat_Vp _
⬝ !concat_1p)
-- The function equiv_rect says that given an equivalence f : A → B,
-- and a hypothesis from B, one may always assume that the hypothesis
-- is in the image of e.
-- In fibrational terms, if we have a fibration over B which has a section
-- once pulled back along an equivalence f : A → B, then it has a section
-- over all of B.
definition equiv_rect (P : B -> Type) :
(Πx, P (f x)) → (Πy, P y) :=
(λg y, path.transport _ (retr f y) (g (f⁻¹ y)))
definition equiv_rect_comp (P : B → Type)
(df : Π (x : A), P (f x)) (x : A) : equiv_rect f P df (f x) ≈ df x :=
let eq1 := (apD df (sect f x)) in
calc equiv_rect f P df (f x)
≈ transport P (retr f (f x)) (df (f⁻¹ (f x))) : idp
... ≈ transport P (ap f (sect f x)) (df (f⁻¹ (f x))) : adj f
... ≈ transport (P ∘ f) (sect f x) (df (f⁻¹ (f x))) : transport_compose
... ≈ df x : eq1
end
--Transporting is an equivalence
definition transport [instance] (P : A → Type) {x y : A} (p : x ≈ y) : (IsEquiv (transport P p)) :=
IsEquiv.mk (transport P (p⁻¹)) (transport_pV P p) (transport_Vp P p) (transport_pVp P p)
end IsEquiv
namespace Equiv
context
parameters {A B C : Type} (eqf : A ≃ B)
private definition f : A → B := equiv_fun eqf
private definition Hf : IsEquiv f := equiv_isequiv eqf
protected definition id : A ≃ A := Equiv.mk id IsEquiv.id_closed
theorem compose (eqg: B ≃ C) : A ≃ C :=
Equiv.mk ((equiv_fun eqg) ∘ f)
(IsEquiv.comp_closed Hf (equiv_isequiv eqg))
theorem path_closed (f' : A → B) (Heq : equiv_fun eqf ≈ f') : A ≃ B :=
Equiv.mk f' (IsEquiv.path_closed Hf Heq)
theorem inv_closed : B ≃ A :=
Equiv.mk (IsEquiv.inv f) !IsEquiv.inv_closed
theorem cancel_R {g : B → C} (Hgf : IsEquiv (g ∘ f)) : B ≃ C :=
Equiv.mk g (IsEquiv.cancel_R f _)
theorem cancel_L {g : C → A} (Hgf : IsEquiv (f ∘ g)) : C ≃ A :=
Equiv.mk g (IsEquiv.cancel_L f _)
theorem transport (P : A → Type) {x y : A} {p : x ≈ y} : (P x) ≃ (P y) :=
Equiv.mk (transport P p) (IsEquiv.transport P p)
end
-- calc enviroment
-- Note: Calculating with substitutions needs univalence
calc_trans compose
calc_refl id
calc_symm inv_closed
end Equiv