lean2/hott/init/equiv.hlean

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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
-/
prelude
import .path .function
open eq function lift
/- 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) :=
mk' ::
(inv : B → A)
(right_inv : Πb, f (inv b) = b)
(left_inv : Πa, inv (f a) = a)
(adj : Πx, right_inv (f x) = ap f (left_inv x))
attribute is_equiv.inv [quasireducible]
-- A more bundled version of equivalence
structure equiv (A B : Type) :=
(to_fun : A → B)
(to_is_equiv : is_equiv to_fun)
namespace is_equiv
/- Some instances and closure properties of equivalences -/
postfix `⁻¹` := inv
/- a second notation for the inverse, which is not overloaded -/
postfix [parsing-only] `⁻¹ᶠ`:std.prec.max_plus := inv
section
variables {A B C : Type} (f : A → B) (g : B → C) {f' : A → B}
-- The variant of mk' where f is explicit.
protected abbreviation mk [constructor] := @is_equiv.mk' A B f
-- The identity function is an equivalence.
definition is_equiv_id (A : Type) : (is_equiv (id : A → A)) :=
is_equiv.mk id id (λa, idp) (λa, idp) (λa, idp)
-- The composition of two equivalences is, again, an equivalence.
definition is_equiv_compose [constructor] [Hf : is_equiv f] [Hg : is_equiv g]
: is_equiv (g ∘ f) :=
is_equiv.mk (g ∘ f) (f⁻¹ ∘ g⁻¹)
(λc, ap g (right_inv f (g⁻¹ c)) ⬝ right_inv g c)
(λa, ap (inv f) (left_inv g (f a)) ⬝ left_inv f a)
(λa, (whisker_left _ (adj g (f a))) ⬝
(ap_con g _ _)⁻¹ ⬝
ap02 g ((ap_con_eq_con (right_inv f) (left_inv g (f a)))⁻¹ ⬝
(ap_compose f (inv f) _ ◾ adj f a) ⬝
(ap_con f _ _)⁻¹
) ⬝
(ap_compose g f _)⁻¹
)
-- Any function equal to an equivalence is an equivlance as well.
variable {f}
definition is_equiv_eq_closed [Hf : is_equiv f] (Heq : f = f') : is_equiv f' :=
eq.rec_on Heq Hf
end
section
parameters {A B : Type} (f : A → B) (g : B → A)
(ret : Πb, f (g b) = b) (sec : Πa, g (f a) = a)
private definition adjointify_left_inv' (a : A) : g (f a) = a :=
ap g (ap f (inverse (sec a))) ⬝ ap g (ret (f a)) ⬝ sec a
private definition adjointify_adj' (a : A) : ret (f a) = ap f (adjointify_left_inv' 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) : by rewrite idp_con
... = (ret (f a) ⬝ (ret (f a))⁻¹) ⬝ ap f (sec a) : by rewrite con.right_inv
... = ((ret fgfa)⁻¹ ⬝ ap (f ∘ g) (ret (f a))) ⬝ ap f (sec a) : by rewrite con_ap_eq_con
... = ((ret fgfa)⁻¹ ⬝ fgretrfa) ⬝ ap f (sec a) : by rewrite ap_compose
... = (ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a)) : by rewrite con.assoc,
have eq2 : ap f (sec a) ⬝ idp = (ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a)),
from !con_idp ⬝ eq1,
have eq3 : idp = _,
from calc idp
= (ap f (sec a))⁻¹ ⬝ ((ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a))) : eq_inv_con_of_con_eq eq2
... = ((ap f (sec a))⁻¹ ⬝ (ret fgfa)⁻¹) ⬝ (fgretrfa ⬝ ap f (sec a)) : by rewrite con.assoc'
... = (ap f (sec a)⁻¹ ⬝ (ret fgfa)⁻¹) ⬝ (fgretrfa ⬝ ap f (sec a)) : by rewrite ap_inv
... = ((ap f (sec a)⁻¹ ⬝ (ret fgfa)⁻¹) ⬝ fgretrfa) ⬝ ap f (sec a) : by rewrite con.assoc'
... = ((retrfa⁻¹ ⬝ ap (f ∘ g) (ap f (sec a)⁻¹)) ⬝ fgretrfa) ⬝ ap f (sec a) : by rewrite con_ap_eq_con
... = ((retrfa⁻¹ ⬝ fgfinvsect) ⬝ fgretrfa) ⬝ ap f (sec a) : by rewrite ap_compose
... = (retrfa⁻¹ ⬝ (fgfinvsect ⬝ fgretrfa)) ⬝ ap f (sec a) : by rewrite con.assoc'
... = retrfa⁻¹ ⬝ ap f (ap g (ap f (sec a)⁻¹) ⬝ ap g (ret (f a))) ⬝ ap f (sec a) : by rewrite ap_con
... = retrfa⁻¹ ⬝ (ap f (ap g (ap f (sec a)⁻¹) ⬝ ap g (ret (f a))) ⬝ ap f (sec a)) : by rewrite con.assoc'
... = retrfa⁻¹ ⬝ ap f ((ap g (ap f (sec a)⁻¹) ⬝ ap g (ret (f a))) ⬝ sec a) : by rewrite -ap_con,
have eq4 : ret (f a) = ap f ((ap g (ap f (sec a)⁻¹) ⬝ ap g (ret (f a))) ⬝ sec a),
from eq_of_idp_eq_inv_con eq3,
eq4
definition adjointify [constructor] : is_equiv f :=
is_equiv.mk f g ret adjointify_left_inv' adjointify_adj'
end
-- Any function pointwise equal to an equivalence is an equivalence as well.
definition homotopy_closed [constructor] {A B : Type} (f : A → B) {f' : A → B} [Hf : is_equiv f]
(Hty : f ~ f') : is_equiv f' :=
adjointify f'
(inv f)
(λ b, (Hty (inv f b))⁻¹ ⬝ right_inv f b)
(λ a, (ap (inv f) (Hty a))⁻¹ ⬝ left_inv f a)
definition inv_homotopy_closed [constructor] {A B : Type} {f : A → B} {f' : B → A}
[Hf : is_equiv f] (Hty : f⁻¹ ~ f') : is_equiv f :=
adjointify f
f'
(λ b, ap f !Hty⁻¹ ⬝ right_inv f b)
(λ a, !Hty⁻¹ ⬝ left_inv f a)
definition is_equiv_up [instance] [constructor] (A : Type) : is_equiv (up : A → lift A) :=
adjointify up down (λa, by induction a;reflexivity) (λa, idp)
section
variables {A B C : Type} (f : A → B) {f' : A → B} [Hf : is_equiv f] (g : B → C)
include Hf
--The inverse of an equivalence is, again, an equivalence.
definition is_equiv_inv [instance] [constructor] : is_equiv f⁻¹ :=
adjointify f⁻¹ f (left_inv f) (right_inv f)
definition cancel_right (g : B → C) [Hgf : is_equiv (g ∘ f)] : (is_equiv g) :=
have Hfinv [visible] : is_equiv f⁻¹, from is_equiv_inv f,
@homotopy_closed _ _ _ _ (is_equiv_compose f⁻¹ (g ∘ f)) (λb, ap g (@right_inv _ _ f _ b))
definition cancel_left (g : C → A) [Hgf : is_equiv (f ∘ g)] : (is_equiv g) :=
have Hfinv [visible] : is_equiv f⁻¹, from is_equiv_inv f,
@homotopy_closed _ _ _ _ (is_equiv_compose (f ∘ g) f⁻¹) (λa, left_inv f (g a))
definition eq_of_fn_eq_fn' {x y : A} (q : f x = f y) : x = y :=
(left_inv f x)⁻¹ ⬝ ap f⁻¹ q ⬝ left_inv f y
definition is_equiv_ap [instance] (x y : A) : is_equiv (ap f : x = y → f x = f y) :=
adjointify
(ap f)
(eq_of_fn_eq_fn' f)
(λq, !ap_con
⬝ whisker_right !ap_con _
⬝ ((!ap_inv ⬝ inverse2 (adj f _)⁻¹)
◾ (inverse (ap_compose f f⁻¹ _))
◾ (adj f _)⁻¹)
⬝ con_ap_con_eq_con_con (right_inv f) _ _
⬝ whisker_right !con.left_inv _
⬝ !idp_con)
(λp, whisker_right (whisker_left _ (ap_compose f⁻¹ f _)⁻¹) _
⬝ con_ap_con_eq_con_con (left_inv f) _ _
⬝ whisker_right !con.left_inv _
⬝ !idp_con)
-- 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 is_equiv_rect (P : B → Type) :
(Πx, P (f x)) → (Πy, P y) :=
(λg y, eq.transport _ (right_inv f y) (g (f⁻¹ y)))
definition is_equiv_rect_comp (P : B → Type)
(df : Π (x : A), P (f x)) (x : A) : is_equiv_rect f P df (f x) = df x :=
calc
is_equiv_rect f P df (f x)
= right_inv f (f x) ▸ df (f⁻¹ (f x)) : by esimp
... = ap f (left_inv f x) ▸ df (f⁻¹ (f x)) : by rewrite -adj
... = left_inv f x ▸ df (f⁻¹ (f x)) : by rewrite -tr_compose
... = df x : by rewrite (apd df (left_inv f x))
end
section
variables {A B : Type} {f : A → B} [Hf : is_equiv f] {a : A} {b : B}
include Hf
--Rewrite rules
definition eq_of_eq_inv (p : a = f⁻¹ b) : f a = b :=
ap f p ⬝ right_inv f b
definition eq_of_inv_eq (p : f⁻¹ b = a) : b = f a :=
(eq_of_eq_inv p⁻¹)⁻¹
definition inv_eq_of_eq (p : b = f a) : f⁻¹ b = a :=
ap f⁻¹ p ⬝ left_inv f a
definition eq_inv_of_eq (p : f a = b) : a = f⁻¹ b :=
(inv_eq_of_eq p⁻¹)⁻¹
end
--Transporting is an equivalence
definition is_equiv_tr [instance] [constructor] {A : Type} (P : A → Type) {x y : A} (p : x = y)
: (is_equiv (transport P p)) :=
is_equiv.mk _ (transport P p⁻¹) (tr_inv_tr p) (inv_tr_tr p) (tr_inv_tr_lemma p)
end is_equiv
open is_equiv
namespace eq
definition tr_inv_fn {A : Type} {B : A → Type} {a a' : A} (p : a = a') :
transport B p⁻¹ = (transport B p)⁻¹ := idp
definition tr_inv {A : Type} {B : A → Type} {a a' : A} (p : a = a') (b : B a') :
p⁻¹ ▸ b = (transport B p)⁻¹ b := idp
definition cast_inv_fn {A B : Type} (p : A = B) : cast p⁻¹ = (cast p)⁻¹ := idp
definition cast_inv {A B : Type} (p : A = B) (b : B) : cast p⁻¹ b = (cast p)⁻¹ b := idp
end eq
namespace equiv
namespace ops
attribute to_fun [coercion]
end ops
open equiv.ops
attribute to_is_equiv [instance]
infix `≃`:25 := equiv
variables {A B C : Type}
protected definition MK [reducible] [constructor] (f : A → B) (g : B → A)
(right_inv : Πb, f (g b) = b) (left_inv : Πa, g (f a) = a) : A ≃ B :=
equiv.mk f (adjointify f g right_inv left_inv)
definition to_inv [reducible] [unfold 3] (f : A ≃ B) : B → A := f⁻¹
definition to_right_inv [reducible] [unfold 3] (f : A ≃ B) (b : B) : f (f⁻¹ b) = b :=
right_inv f b
definition to_left_inv [reducible] [unfold 3] (f : A ≃ B) (a : A) : f⁻¹ (f a) = a :=
left_inv f a
protected definition refl [refl] [constructor] : A ≃ A :=
equiv.mk id !is_equiv_id
protected definition symm [symm] (f : A ≃ B) : B ≃ A :=
equiv.mk f⁻¹ !is_equiv_inv
protected definition trans [trans] (f : A ≃ B) (g : B ≃ C) : A ≃ C :=
equiv.mk (g ∘ f) !is_equiv_compose
infixl `⬝e`:75 := equiv.trans
postfix [parsing-only] `⁻¹ᵉ`:(max + 1) := equiv.symm
-- notation for inverse which is not overloaded
abbreviation erfl [constructor] := @equiv.refl
definition to_inv_trans [reducible] [unfold-full] (f : A ≃ B) (g : B ≃ C)
: to_inv (f ⬝e g) = to_fun (g⁻¹ᵉ ⬝e f⁻¹ᵉ) :=
idp
definition equiv_change_fun [constructor] (f : A ≃ B) {f' : A → B} (Heq : f ~ f') : A ≃ B :=
equiv.mk f' (is_equiv.homotopy_closed f Heq)
definition equiv_change_inv [constructor] (f : A ≃ B) {f' : B → A} (Heq : f⁻¹ ~ f')
: A ≃ B :=
equiv.mk f (inv_homotopy_closed Heq)
--rename: eq_equiv_fn_eq_of_is_equiv
definition eq_equiv_fn_eq [constructor] (f : A → B) [H : is_equiv f] (a b : A) : (a = b) ≃ (f a = f b) :=
equiv.mk (ap f) !is_equiv_ap
--rename: eq_equiv_fn_eq
definition eq_equiv_fn_eq_of_equiv [constructor] (f : A ≃ B) (a b : A) : (a = b) ≃ (f a = f b) :=
equiv.mk (ap f) !is_equiv_ap
definition equiv_ap (P : A → Type) {a b : A} (p : a = b) : (P a) ≃ (P b) :=
equiv.mk (transport P p) !is_equiv_tr
definition eq_of_fn_eq_fn (f : A ≃ B) {x y : A} (q : f x = f y) : x = y :=
(left_inv f x)⁻¹ ⬝ ap f⁻¹ q ⬝ left_inv f y
definition eq_of_fn_eq_fn_inv (f : A ≃ B) {x y : B} (q : f⁻¹ x = f⁻¹ y) : x = y :=
(right_inv f x)⁻¹ ⬝ ap f q ⬝ right_inv f y
--we need this theorem for the funext_of_ua proof
theorem inv_eq {A B : Type} (eqf eqg : A ≃ B) (p : eqf = eqg) : (to_fun eqf)⁻¹ = (to_fun eqg)⁻¹ :=
eq.rec_on p idp
definition equiv_of_equiv_of_eq [trans] {A B C : Type} (p : A = B) (q : B ≃ C) : A ≃ C := p⁻¹ ▸ q
definition equiv_of_eq_of_equiv [trans] {A B C : Type} (p : A ≃ B) (q : B = C) : A ≃ C := q ▸ p
definition equiv_lift (A : Type) : A ≃ lift A := equiv.mk up _
definition equiv_rect (f : A ≃ B) (P : B → Type) :
(Πx, P (f x)) → (Πy, P y) :=
(λg y, eq.transport _ (right_inv f y) (g (f⁻¹ y)))
definition equiv_rect_comp (f : A ≃ B) (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)
= right_inv f (f x) ▸ df (f⁻¹ (f x)) : by esimp
... = ap f (left_inv f x) ▸ df (f⁻¹ (f x)) : by rewrite -adj
... = left_inv f x ▸ df (f⁻¹ (f x)) : by rewrite -tr_compose
... = df x : by rewrite (apd df (left_inv f x))
namespace ops
postfix `⁻¹` := equiv.symm -- overloaded notation for inverse
end ops
end equiv
export [unfold-hints] equiv [unfold-hints] is_equiv