lean2/hott/init/equiv.hlean
Leonardo de Moura 4f2e0c6d7f refactor(frontends/lean): add 'attribute' command
The new command provides a uniform way to set declaration attributes.
It replaces the commands: class, instance, coercion, multiple_instances,
reducible, irreducible
2015-01-24 20:23:21 -08:00

267 lines
11 KiB
Text
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

-- 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
prelude
import .path .function
open eq function
-- Equivalences
-- ------------
-- This is our definition of equivalence. In the HoTT-book it's called
-- ihae (half-adjoint equivalence).
structure is_equiv [class] {A B : Type} (f : A → B) :=
(inv : B → A)
(retr : (f ∘ inv) id)
(sect : (inv ∘ f) id)
(adj : Πx, retr (f x) = ap f (sect x))
-- A more bundled version of equivalence to calculate with
structure equiv (A B : Type) :=
(to_fun : A → B)
(to_is_equiv : is_equiv to_fun)
-- Some instances and closure properties of equivalences
namespace is_equiv
postfix `⁻¹` := inv
variables {A B C : Type} (f : A → B) (g : B → C) {f' : A → B}
-- The identity function is an equivalence.
definition id_is_equiv : (@is_equiv A A id) := is_equiv.mk id (λa, idp) (λa, idp) (λa, idp)
-- The composition of two equivalences is, again, an equivalence.
protected definition compose [Hf : is_equiv f] [Hg : is_equiv g] : (is_equiv (g ∘ f)) :=
is_equiv.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 : is_equiv f] (Heq : f = f') : (is_equiv f') :=
eq.rec_on Heq Hf
-- Any function pointwise equal to an equivalence is an equivalence as well.
definition homotopy_closed [Hf : is_equiv f] (Hty : f f') : (is_equiv f') :=
let sect' := (λ b, (Hty (inv f b))⁻¹ ⬝ retr f b) in
let retr' := (λ a, (ap (inv f) (Hty a))⁻¹ ⬝ sect f a) in
let adj' := (λ (a : A),
let ff'a := Hty 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 _ _)
... = ap (f ∘ invf) ff'a ⬝ retrf'a : concat_A1p
... = ap f (ap invf ff'a) ⬝ retrf'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 ⬝ Hty a) : ap_V invf ff'a
... = ap f (ap invf ff'a)⁻¹ ⬝ (Hty (invf (f a)) ⬝ ap f' secta) : concat_Ap
... = (ap f (ap invf ff'a)⁻¹ ⬝ Hty (invf (f a))) ⬝ ap f' secta : concat_pp_p
... = (ap f ((ap invf ff'a)⁻¹) ⬝ Hty (invf (f a))) ⬝ ap f' secta : ap_V
... = (Hty (invf (f' a)) ⬝ ap f' ((ap invf ff'a)⁻¹)) ⬝ ap f' secta : concat_Ap
... = (Hty (invf (f' a)) ⬝ (ap f' (ap invf ff'a))⁻¹) ⬝ ap f' secta : ap_V
... = Hty (invf (f' a)) ⬝ ((ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta) : concat_pp_p,
have eq3 : _ = _,
from calc (Hty (invf (f' a)))⁻¹ ⬝ retrf'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
... = ap f' ((ap invf ff'a)⁻¹ ⬝ secta) : ap_pp,
eq3) in
is_equiv.mk (inv f) sect' retr' adj'
end is_equiv
namespace is_equiv
context
parameters {A B : Type} (f : A → B) (g : B → A)
(ret : f ∘ g id) (sec : g ∘ f id)
definition adjointify_sect' : g ∘ f id :=
(λx, ap g (ap f (inverse (sec x))) ⬝ ap g (ret (f x)) ⬝ sec x)
definition adjointify_adj' : Π (x : A), ret (f x) = ap f (adjointify_sect' x) :=
(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)
definition adjointify : is_equiv f :=
is_equiv.mk g ret adjointify_sect' adjointify_adj'
end
end is_equiv
namespace is_equiv
variables {A B: Type} (f : A → B)
--The inverse of an equivalence is, again, an equivalence.
definition inv_closed [instance] [Hf : is_equiv f] : (is_equiv (inv f)) :=
adjointify (inv f) f (sect f) (retr f)
end is_equiv
namespace is_equiv
variables {A : Type}
section
variables {B C : Type} (f : A → B) {f' : A → B} [Hf : is_equiv f]
include Hf
definition cancel_R (g : B → C) [Hgf : is_equiv (g ∘ f)] : (is_equiv g) :=
have Hfinv [visible] : is_equiv (f⁻¹), from inv_closed f,
@homotopy_closed _ _ _ _ (compose (f⁻¹) (g ∘ f)) (λb, ap g (@retr _ _ f _ b))
definition cancel_L (g : C → A) [Hgf : is_equiv (f ∘ g)] : (is_equiv g) :=
have Hfinv [visible] : is_equiv (f⁻¹), from inv_closed f,
@homotopy_closed _ _ _ _ (compose (f ∘ g) (f⁻¹)) (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 (f⁻¹) 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) : is_equiv (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, eq.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 :=
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 : apD df (sect f x)
end
--Transporting is an equivalence
protected definition transport [instance] (P : A → Type) {x y : A} (p : x = y) : (is_equiv (transport P p)) :=
is_equiv.mk (transport P (p⁻¹)) (transport_pV P p) (transport_Vp P p) (transport_pVp P p)
end is_equiv
namespace equiv
persistent attribute to_is_equiv [instance]
infix `≃`:25 := equiv
context
parameters {A B C : Type} (eqf : A ≃ B)
private definition f : A → B := to_fun eqf
private definition Hf [instance] : is_equiv f := to_is_equiv eqf
protected definition refl : A ≃ A := equiv.mk id is_equiv.id_is_equiv
theorem trans (eqg: B ≃ C) : A ≃ C :=
equiv.mk ((to_fun eqg) ∘ f)
(is_equiv.compose f (to_fun eqg))
theorem path_closed (f' : A → B) (Heq : to_fun eqf = f') : A ≃ B :=
equiv.mk f' (is_equiv.path_closed f Heq)
theorem symm : B ≃ A :=
equiv.mk (is_equiv.inv f) !is_equiv.inv_closed
theorem cancel_R (g : B → C) [Hgf : is_equiv (g ∘ f)] : B ≃ C :=
equiv.mk g (is_equiv.cancel_R f _)
theorem cancel_L (g : C → A) [Hgf : is_equiv (f ∘ g)] : C ≃ A :=
equiv.mk g (is_equiv.cancel_L f _)
protected theorem transport (P : A → Type) {x y : A} {p : x = y} : (P x) ≃ (P y) :=
equiv.mk (transport P p) (is_equiv.transport P p)
end
context
parameters {A B : Type} (eqf eqg : A ≃ B)
private definition Hf [instance] : is_equiv (to_fun eqf) := to_is_equiv eqf
private definition Hg [instance] : is_equiv (to_fun eqg) := to_is_equiv eqg
--We need this theorem for the funext_from_ua proof
theorem inv_eq (p : eqf = eqg)
: is_equiv.inv (to_fun eqf) = is_equiv.inv (to_fun eqg) :=
eq.rec_on p idp
end
-- calc enviroment
-- Note: Calculating with substitutions needs univalence
calc_trans trans
calc_refl refl
calc_symm symm
end equiv