619 lines
24 KiB
Text
619 lines
24 KiB
Text
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||
-- Author: Jeremy Avigad
|
||
-- Ported from Coq HoTT
|
||
|
||
notation `assume` binders `,` r:(scoped f, f) := r
|
||
notation `take` binders `,` r:(scoped f, f) := r
|
||
|
||
definition id {A : Type} (a : A) := a
|
||
definition compose {A : Type} {B : Type} {C : Type} (g : B → C) (f : A → B) := λ x, g (f x)
|
||
infixl `∘`:60 := compose
|
||
|
||
-- Path
|
||
-- ----
|
||
|
||
inductive path {A : Type} (a : A) : A → Type :=
|
||
idpath : path a a
|
||
definition idpath := @path.idpath
|
||
infix `≈`:50 := path
|
||
-- TODO: is this right?
|
||
notation x `≈`:50 y `:>`:0 A:0 := @path A x y
|
||
notation `idp`:max := idpath _ -- TODO: can we / should we use `1`?
|
||
|
||
namespace path
|
||
-- Concatenation and inverse
|
||
-- -------------------------
|
||
|
||
definition concat {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) : x ≈ z :=
|
||
path.rec (λu, u) q p
|
||
|
||
definition inverse {A : Type} {x y : A} (p : x ≈ y) : y ≈ x :=
|
||
path.rec (idpath x) p
|
||
|
||
infixl `⬝`:75 := concat
|
||
postfix `^`:100 := inverse
|
||
|
||
-- In Coq, these are not needed, because concat and inv are kept transparent
|
||
definition inv_1 {A : Type} (x : A) : (idpath x)^ ≈ idpath x := idp
|
||
|
||
definition concat_11 {A : Type} (x : A) : idpath x ⬝ idpath x ≈ idpath x := idp
|
||
|
||
|
||
-- The 1-dimensional groupoid structure
|
||
-- ------------------------------------
|
||
|
||
-- The identity path is a right unit.
|
||
definition concat_p1 {A : Type} {x y : A} (p : x ≈ y) : p ⬝ idp ≈ p :=
|
||
rec_on p idp
|
||
|
||
-- The identity path is a right unit.
|
||
definition concat_1p {A : Type} {x y : A} (p : x ≈ y) : idp ⬝ p ≈ p :=
|
||
rec_on p idp
|
||
|
||
-- Concatenation is associative.
|
||
definition concat_p_pp {A : Type} {x y z t : A} (p : x ≈ y) (q : y ≈ z) (r : z ≈ t) :
|
||
p ⬝ (q ⬝ r) ≈ (p ⬝ q) ⬝ r :=
|
||
rec_on r (rec_on q idp)
|
||
|
||
definition concat_pp_p {A : Type} {x y z t : A} (p : x ≈ y) (q : y ≈ z) (r : z ≈ t) :
|
||
(p ⬝ q) ⬝ r ≈ p ⬝ (q ⬝ r) :=
|
||
rec_on r (rec_on q idp)
|
||
|
||
-- The left inverse law.
|
||
definition concat_pV {A : Type} {x y : A} (p : x ≈ y) : p ⬝ p^ ≈ idp :=
|
||
rec_on p idp
|
||
|
||
-- The right inverse law.
|
||
definition concat_Vp {A : Type} {x y : A} (p : x ≈ y) : p^ ⬝ p ≈ idp :=
|
||
rec_on p idp
|
||
|
||
|
||
-- Several auxiliary theorems about canceling inverses across associativity. These are somewhat
|
||
-- redundant, following from earlier theorems.
|
||
|
||
definition concat_V_pp {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) : p^ ⬝ (p ⬝ q) ≈ q :=
|
||
rec_on q (rec_on p idp)
|
||
|
||
definition concat_p_Vp {A : Type} {x y z : A} (p : x ≈ y) (q : x ≈ z) : p ⬝ (p^ ⬝ q) ≈ q :=
|
||
rec_on q (rec_on p idp)
|
||
|
||
definition concat_pp_V {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) : (p ⬝ q) ⬝ q^ ≈ p :=
|
||
rec_on q (rec_on p idp)
|
||
|
||
definition concat_pV_p {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) : (p ⬝ q^) ⬝ q ≈ p :=
|
||
rec_on q (take p, rec_on p idp) p
|
||
|
||
-- Inverse distributes over concatenation
|
||
definition inv_pp {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) : (p ⬝ q)^ ≈ q^ ⬝ p^ :=
|
||
rec_on q (rec_on p idp)
|
||
|
||
definition inv_Vp {A : Type} {x y z : A} (p : y ≈ x) (q : y ≈ z) : (p^ ⬝ q)^ ≈ q^ ⬝ p :=
|
||
rec_on q (rec_on p idp)
|
||
|
||
-- universe metavariables
|
||
definition inv_pV {A : Type} {x y z : A} (p : x ≈ y) (q : z ≈ y) : (p ⬝ q^)^ ≈ q ⬝ p^ :=
|
||
rec_on p (λq, rec_on q idp) q
|
||
|
||
definition inv_VV {A : Type} {x y z : A} (p : y ≈ x) (q : z ≈ y) : (p^ ⬝ q^)^ ≈ q ⬝ p :=
|
||
rec_on p (rec_on q idp)
|
||
|
||
-- Inverse is an involution.
|
||
definition inv_V {A : Type} {x y : A} (p : x ≈ y) : p^^ ≈ p :=
|
||
rec_on p idp
|
||
|
||
|
||
-- Theorems for moving things around in equations
|
||
-- ----------------------------------------------
|
||
|
||
definition moveR_Mp {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : y ≈ x) :
|
||
p ≈ (r^ ⬝ q) → (r ⬝ p) ≈ q :=
|
||
have gen : Πp q, p ≈ (r^ ⬝ q) → (r ⬝ p) ≈ q, from
|
||
rec_on r
|
||
(take p q,
|
||
assume h : p ≈ idp^ ⬝ q,
|
||
show idp ⬝ p ≈ q, from concat_1p _ ⬝ h ⬝ concat_1p _),
|
||
gen p q
|
||
|
||
definition moveR_pM {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : y ≈ x) :
|
||
r ≈ q ⬝ p^ → r ⬝ p ≈ q :=
|
||
rec_on p (take q r h, (concat_p1 _ ⬝ h ⬝ concat_p1 _)) q r
|
||
|
||
definition moveR_Vp {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : x ≈ y) :
|
||
p ≈ r ⬝ q → r^ ⬝ p ≈ q :=
|
||
rec_on r (take p q h, concat_1p _ ⬝ h ⬝ concat_1p _) p q
|
||
|
||
definition moveR_pV {A : Type} {x y z : A} (p : z ≈ x) (q : y ≈ z) (r : y ≈ x) :
|
||
r ≈ q ⬝ p → r ⬝ p^ ≈ q :=
|
||
rec_on p (take q r h, concat_p1 _ ⬝ h ⬝ concat_p1 _) q r
|
||
|
||
definition moveL_Mp {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : y ≈ x) :
|
||
r^ ⬝ q ≈ p → q ≈ r ⬝ p :=
|
||
rec_on r (take p q h, (concat_1p _)^ ⬝ h ⬝ (concat_1p _)^) p q
|
||
|
||
definition moveL_pM {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : y ≈ x) :
|
||
q ⬝ p^ ≈ r → q ≈ r ⬝ p :=
|
||
rec_on p (take q r h, (concat_p1 _)^ ⬝ h ⬝ (concat_p1 _)^) q r
|
||
|
||
definition moveL_Vp {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : x ≈ y) :
|
||
r ⬝ q ≈ p → q ≈ r^ ⬝ p :=
|
||
rec_on r (take p q h, (concat_1p _)^ ⬝ h ⬝ (concat_1p _)^) p q
|
||
|
||
definition moveL_pV {A : Type} {x y z : A} (p : z ≈ x) (q : y ≈ z) (r : y ≈ x) :
|
||
q ⬝ p ≈ r → q ≈ r ⬝ p^ :=
|
||
rec_on p (take q r h, (concat_p1 _)^ ⬝ h ⬝ (concat_p1 _)^) q r
|
||
|
||
definition moveL_1M {A : Type} {x y : A} (p q : x ≈ y) :
|
||
p ⬝ q^ ≈ idp → p ≈ q :=
|
||
rec_on q (take p h, (concat_p1 _)^ ⬝ h) p
|
||
|
||
definition moveL_M1 {A : Type} {x y : A} (p q : x ≈ y) :
|
||
q^ ⬝ p ≈ idp → p ≈ q :=
|
||
rec_on q (take p h, (concat_1p _)^ ⬝ h) p
|
||
|
||
definition moveL_1V {A : Type} {x y : A} (p : x ≈ y) (q : y ≈ x) :
|
||
p ⬝ q ≈ idp → p ≈ q^ :=
|
||
rec_on q (take p h, (concat_p1 _)^ ⬝ h) p
|
||
|
||
definition moveL_V1 {A : Type} {x y : A} (p : x ≈ y) (q : y ≈ x) :
|
||
q ⬝ p ≈ idp → p ≈ q^ :=
|
||
rec_on q (take p h, (concat_1p _)^ ⬝ h) p
|
||
|
||
definition moveR_M1 {A : Type} {x y : A} (p q : x ≈ y) :
|
||
idp ≈ p^ ⬝ q → p ≈ q :=
|
||
rec_on p (take q h, h ⬝ (concat_1p _)) q
|
||
|
||
definition moveR_1M {A : Type} {x y : A} (p q : x ≈ y) :
|
||
idp ≈ q ⬝ p^ → p ≈ q :=
|
||
rec_on p (take q h, h ⬝ (concat_p1 _)) q
|
||
|
||
definition moveR_1V {A : Type} {x y : A} (p : x ≈ y) (q : y ≈ x) :
|
||
idp ≈ q ⬝ p → p^ ≈ q :=
|
||
rec_on p (take q h, h ⬝ (concat_p1 _)) q
|
||
|
||
definition moveR_V1 {A : Type} {x y : A} (p : x ≈ y) (q : y ≈ x) :
|
||
idp ≈ p ⬝ q → p^ ≈ q :=
|
||
rec_on p (take q h, h ⬝ (concat_1p _)) q
|
||
|
||
|
||
-- Transport
|
||
-- ---------
|
||
|
||
-- keep transparent, so transport _ idp p is definitionally equal to p
|
||
definition transport {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (u : P x) : P y :=
|
||
path.rec_on p u
|
||
|
||
definition transport_1 {A : Type} (P : A → Type) {x : A} (u : P x) : transport _ idp u ≈ u :=
|
||
idp
|
||
|
||
-- TODO: is the binding strength on x reasonable? (It is modeled on the notation for subst
|
||
-- in the standard library.)
|
||
-- This idiom makes the operation right associative.
|
||
notation p `#`:65 x:64 := transport _ p x
|
||
|
||
definition ap ⦃A B : Type⦄ (f : A → B) {x y:A} (p : x ≈ y) : f x ≈ f y :=
|
||
path.rec_on p idp
|
||
|
||
-- TODO: is this better than an alias? Note use of curly brackets
|
||
definition ap01 := ap
|
||
|
||
definition pointwise_paths {A : Type} {P : A → Type} (f g : Πx, P x) : Type :=
|
||
Πx : A, f x ≈ g x
|
||
|
||
infix `∼`:50 := pointwise_paths
|
||
|
||
definition apD10 {A} {B : A → Type} {f g : Πx, B x} (H : f ≈ g) : f ∼ g :=
|
||
λx, path.rec_on H idp
|
||
|
||
definition ap10 {A B} {f g : A → B} (H : f ≈ g) : f ∼ g := apD10 H
|
||
|
||
definition ap11 {A B} {f g : A → B} (H : f ≈ g) {x y : A} (p : x ≈ y) : f x ≈ g y :=
|
||
rec_on H (rec_on p idp)
|
||
|
||
-- TODO: Note that the next line breaks the proof!
|
||
-- opaque_hint (hiding rec_on)
|
||
-- set_option pp.implicit true
|
||
definition apD {A:Type} {B : A → Type} (f : Πa:A, B a) {x y : A} (p : x ≈ y) : p # (f x) ≈ f y :=
|
||
rec_on p idp
|
||
|
||
|
||
-- More theorems for moving things around in equations
|
||
-- ---------------------------------------------------
|
||
|
||
definition moveR_transport_p {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (u : P x) (v : P y) :
|
||
u ≈ p^ # v → p # u ≈ v :=
|
||
rec_on p (take u v, id) u v
|
||
|
||
definition moveR_transport_V {A : Type} (P : A → Type) {x y : A} (p : y ≈ x) (u : P x) (v : P y) :
|
||
u ≈ p # v → p^ # u ≈ v :=
|
||
rec_on p (take u v, id) u v
|
||
|
||
definition moveL_transport_V {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (u : P x) (v : P y) :
|
||
p # u ≈ v → u ≈ p^ # v :=
|
||
rec_on p (take u v, id) u v
|
||
|
||
definition moveL_transport_p {A : Type} (P : A → Type) {x y : A} (p : y ≈ x) (u : P x) (v : P y) :
|
||
p^ # u ≈ v → u ≈ p # v :=
|
||
rec_on p (take u v, id) u v
|
||
|
||
|
||
-- Functoriality of functions
|
||
-- --------------------------
|
||
|
||
-- Here we prove that functions behave like functors between groupoids, and that [ap] itself is
|
||
-- functorial.
|
||
|
||
-- Functions take identity paths to identity paths
|
||
definition ap_1 {A B : Type} (x : A) (f : A → B) : (ap f idp) ≈ idp :> (f x ≈ f x) := idp
|
||
|
||
definition apD_1 {A B} (x : A) (f : forall x : A, B x) : apD f idp ≈ idp :> (f x ≈ f x) := idp
|
||
|
||
-- Functions commute with concatenation.
|
||
definition ap_pp {A B : Type} (f : A → B) {x y z : A} (p : x ≈ y) (q : y ≈ z) :
|
||
ap f (p ⬝ q) ≈ (ap f p) ⬝ (ap f q) :=
|
||
rec_on q (rec_on p idp)
|
||
|
||
definition ap_p_pp {A B : Type} (f : A → B) {w x y z : A} (r : f w ≈ f x) (p : x ≈ y) (q : y ≈ z) :
|
||
r ⬝ (ap f (p ⬝ q)) ≈ (r ⬝ ap f p) ⬝ (ap f q) :=
|
||
rec_on p (take r q, rec_on q (concat_p_pp r idp idp)) r q
|
||
|
||
definition ap_pp_p {A B : Type} (f : A → B) {w x y z : A} (p : x ≈ y) (q : y ≈ z) (r : f z ≈ f w) :
|
||
(ap f (p ⬝ q)) ⬝ r ≈ (ap f p) ⬝ (ap f q ⬝ r) :=
|
||
rec_on p (take q, rec_on q (take r, concat_pp_p _ _ _)) q r
|
||
|
||
-- Functions commute with path inverses.
|
||
definition inverse_ap {A B : Type} (f : A → B) {x y : A} (p : x ≈ y) : (ap f p)^ ≈ ap f (p^) :=
|
||
rec_on p idp
|
||
|
||
definition ap_V {A B : Type} (f : A → B) {x y : A} (p : x ≈ y) : ap f (p^) ≈ (ap f p)^ :=
|
||
rec_on p idp
|
||
|
||
-- TODO: rename id to idmap?
|
||
definition ap_idmap {A : Type} {x y : A} (p : x ≈ y) : ap id p ≈ p :=
|
||
rec_on p idp
|
||
|
||
definition ap_compose {A B C : Type} (f : A → B) (g : B → C) {x y : A} (p : x ≈ y) :
|
||
ap (g ∘ f) p ≈ ap g (ap f p) :=
|
||
rec_on p idp
|
||
|
||
-- Sometimes we don't have the actual function [compose].
|
||
definition ap_compose' {A B C : Type} (f : A → B) (g : B → C) {x y : A} (p : x ≈ y) :
|
||
ap (λa, g (f a)) p ≈ ap g (ap f p) :=
|
||
rec_on p idp
|
||
|
||
-- The action of constant maps.
|
||
definition ap_const {A B : Type} {x y : A} (p : x ≈ y) (z : B) :
|
||
ap (λu, z) p ≈ idp :=
|
||
rec_on p idp
|
||
|
||
-- Naturality of [ap].
|
||
definition concat_Ap {A B : Type} {f g : A → B} (p : forall x, f x ≈ g x) {x y : A} (q : x ≈ y) :
|
||
(ap f q) ⬝ (p y) ≈ (p x) ⬝ (ap g q) :=
|
||
rec_on q (concat_1p _ ⬝ (concat_p1 _)^)
|
||
|
||
-- Naturality of [ap] at identity.
|
||
definition concat_A1p {A : Type} {f : A → A} (p : forall x, f x ≈ x) {x y : A} (q : x ≈ y) :
|
||
(ap f q) ⬝ (p y) ≈ (p x) ⬝ q :=
|
||
rec_on q (concat_1p _ ⬝ (concat_p1 _)^)
|
||
|
||
definition concat_pA1 {A : Type} {f : A → A} (p : forall x, x ≈ f x) {x y : A} (q : x ≈ y) :
|
||
(p x) ⬝ (ap f q) ≈ q ⬝ (p y) :=
|
||
rec_on q (concat_p1 _ ⬝ (concat_1p _)^)
|
||
|
||
--TODO: note that the Coq proof for the preceding is
|
||
--
|
||
-- match q as i in (_ ≈ y) return (p x ⬝ ap f i ≈ i ⬝ p y) with
|
||
-- | idpath => concat_p1 _ ⬝ (concat_1p _)^
|
||
-- end.
|
||
--
|
||
-- It is nice that we don't have to give the predicate.
|
||
|
||
-- Naturality with other paths hanging around.
|
||
definition concat_pA_pp {A B : Type} {f g : A → B} (p : forall x, f x ≈ g x)
|
||
{x y : A} (q : x ≈ y)
|
||
{w z : B} (r : w ≈ f x) (s : g y ≈ z) :
|
||
(r ⬝ ap f q) ⬝ (p y ⬝ s) ≈ (r ⬝ p x) ⬝ (ap g q ⬝ s) :=
|
||
rec_on q (take s, rec_on s (take r, idp)) s r
|
||
|
||
-- Action of [apD10] and [ap10] on paths
|
||
-- -------------------------------------
|
||
|
||
-- Application of paths between functions preserves the groupoid structure
|
||
|
||
definition apD10_1 {A} {B : A → Type} (f : Πx, B x) (x : A) : apD10 (idpath f) x ≈ idp := idp
|
||
|
||
definition apD10_pp {A} {B : A → Type} {f f' f'' : Πx, B x} (h : f ≈ f') (h' : f' ≈ f'') (x : A) :
|
||
apD10 (h ⬝ h') x ≈ apD10 h x ⬝ apD10 h' x :=
|
||
rec_on h (take h', rec_on h' idp) h'
|
||
|
||
definition apD10_V {A : Type} {B : A → Type} {f g : Πx : A, B x} (h : f ≈ g) (x : A) :
|
||
apD10 (h^) x ≈ (apD10 h x)^ :=
|
||
rec_on h idp
|
||
|
||
definition ap10_1 {A B} {f : A → B} (x : A) : ap10 (idpath f) x ≈ idp := idp
|
||
|
||
definition ap10_pp {A B} {f f' f'' : A → B} (h : f ≈ f') (h' : f' ≈ f'') (x : A) :
|
||
ap10 (h ⬝ h') x ≈ ap10 h x ⬝ ap10 h' x := apD10_pp h h' x
|
||
|
||
definition ap10_V {A B} {f g : A→B} (h : f ≈ g) (x:A) : ap10 (h^) x ≈ (ap10 h x)^ := apD10_V h x
|
||
|
||
-- [ap10] also behaves nicely on paths produced by [ap]
|
||
definition ap_ap10 {A B C} (f g : A → B) (h : B → C) (p : f ≈ g) (a : A) :
|
||
ap h (ap10 p a) ≈ ap10 (ap (λ f', h ∘ f') p) a:=
|
||
rec_on p idp
|
||
|
||
|
||
-- Transport and the groupoid structure of paths
|
||
-- ---------------------------------------------
|
||
|
||
-- TODO: move from above?
|
||
-- definition transport_1 {A : Type} (P : A → Type) {x : A} (u : P x)
|
||
-- : idp # u ≈ u := idp
|
||
|
||
definition transport_pp {A : Type} (P : A → Type) {x y z : A} (p : x ≈ y) (q : y ≈ z) (u : P x) :
|
||
p ⬝ q # u ≈ q # p # u :=
|
||
rec_on q (rec_on p idp)
|
||
|
||
definition transport_pV {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (z : P y) :
|
||
p # p^ # z ≈ z :=
|
||
(transport_pp P (p^) p z)^ ⬝ ap (λr, transport P r z) (concat_Vp p)
|
||
|
||
definition transport_Vp {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (z : P x) :
|
||
p^ # p # z ≈ z :=
|
||
(transport_pp P p (p^) z)^ ⬝ ap (λr, transport P r z) (concat_pV p)
|
||
|
||
|
||
-----------------------------------------------
|
||
-- *** Examples of difficult induction problems
|
||
-----------------------------------------------
|
||
|
||
theorem double_induction
|
||
{A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z)
|
||
{C : Π(x y z : A), Π(p : x ≈ y), Π(q : y ≈ z), Type}
|
||
(H : C x x x (idpath x) (idpath x)) :
|
||
C x y z p q :=
|
||
rec_on p (take z q, rec_on q H) z q
|
||
|
||
theorem double_induction2
|
||
{A : Type} {x y z : A} (p : x ≈ y) (q : z ≈ y)
|
||
{C : Π(x y z : A), Π(p : x ≈ y), Π(q : z ≈ y), Type}
|
||
(H : C z z z (idpath z) (idpath z)) :
|
||
C x y z p q :=
|
||
rec_on p (take y q, rec_on q H) y q
|
||
|
||
theorem double_induction2'
|
||
{A : Type} {x y z : A} (p : x ≈ y) (q : z ≈ y)
|
||
{C : Π(x y z : A), Π(p : x ≈ y), Π(q : z ≈ y), Type}
|
||
(H : C z z z (idpath z) (idpath z)) : C x y z p q :=
|
||
rec_on p (take y q, rec_on q H) y q
|
||
|
||
theorem triple_induction
|
||
{A : Type} {x y z w : A} (p : x ≈ y) (q : y ≈ z) (r : z ≈ w)
|
||
{C : Π(x y z w : A), Π(p : x ≈ y), Π(q : y ≈ z), Π(r: z ≈ w), Type}
|
||
(H : C x x x x (idpath x) (idpath x) (idpath x)) :
|
||
C x y z w p q r :=
|
||
rec_on p (take z q, rec_on q (take w r, rec_on r H)) z q w r
|
||
|
||
-- try this again
|
||
definition concat_pV_p_new {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) : (p ⬝ q^) ⬝ q ≈ p :=
|
||
double_induction2 p q idp
|
||
|
||
definition transport_p_pp {A : Type} (P : A → Type)
|
||
{x y z w : A} (p : x ≈ y) (q : y ≈ z) (r : z ≈ w) (u : P x) :
|
||
ap (λe, e # u) (concat_p_pp p q r) ⬝ (transport_pp P (p ⬝ q) r u) ⬝
|
||
ap (transport P r) (transport_pp P p q u)
|
||
≈ (transport_pp P p (q ⬝ r) u) ⬝ (transport_pp P q r (p # u))
|
||
:> ((p ⬝ (q ⬝ r)) # u ≈ r # q # p # u) :=
|
||
triple_induction p q r (take u, idp) u
|
||
|
||
-- Here is another coherence lemma for transport.
|
||
definition transport_pVp {A} (P : A → Type) {x y : A} (p : x ≈ y) (z : P x) :
|
||
transport_pV P p (transport P p z) ≈ ap (transport P p) (transport_Vp P p z)
|
||
:= rec_on p idp
|
||
|
||
-- Dependent transport in a doubly dependent type.
|
||
definition transportD {A : Type} (B : A → Type) (C : Π a : A, B a → Type)
|
||
{x1 x2 : A} (p : x1 ≈ x2) (y : B x1) (z : C x1 y) :
|
||
C x2 (p # y) :=
|
||
rec_on p z
|
||
|
||
-- Transporting along higher-dimensional paths
|
||
definition transport2 {A : Type} (P : A → Type) {x y : A} {p q : x ≈ y} (r : p ≈ q) (z : P x) :
|
||
p # z ≈ q # z := ap (λp', p' # z) r
|
||
|
||
-- An alternative definition.
|
||
definition transport2_is_ap10 {A : Type} (Q : A → Type) {x y : A} {p q : x ≈ y} (r : p ≈ q)
|
||
(z : Q x) :
|
||
transport2 Q r z ≈ ap10 (ap (transport Q) r) z :=
|
||
rec_on r idp
|
||
|
||
definition transport2_p2p {A : Type} (P : A → Type) {x y : A} {p1 p2 p3 : x ≈ y}
|
||
(r1 : p1 ≈ p2) (r2 : p2 ≈ p3) (z : P x) :
|
||
transport2 P (r1 ⬝ r2) z ≈ transport2 P r1 z ⬝ transport2 P r2 z :=
|
||
rec_on r1 (rec_on r2 idp)
|
||
|
||
-- TODO: another interesting case
|
||
definition transport2_V {A : Type} (Q : A → Type) {x y : A} {p q : x ≈ y} (r : p ≈ q) (z : Q x) :
|
||
transport2 Q (r^) z ≈ ((transport2 Q r z)^) :=
|
||
-- rec_on r idp -- doesn't work
|
||
rec_on r (idpath (inverse (transport2 Q (idpath p) z)))
|
||
|
||
definition concat_AT {A : Type} (P : A → Type) {x y : A} {p q : x ≈ y} {z w : P x} (r : p ≈ q)
|
||
(s : z ≈ w) :
|
||
ap (transport P p) s ⬝ transport2 P r w ≈ transport2 P r z ⬝ ap (transport P q) s :=
|
||
rec_on r (concat_p1 _ ⬝ (concat_1p _)^)
|
||
|
||
-- TODO (from Coq library): What should this be called?
|
||
definition ap_transport {A} {P Q : A → Type} {x y : A} (p : x ≈ y) (f : Πx, P x → Q x) (z : P x) :
|
||
f y (p # z) ≈ (p # (f x z)) :=
|
||
rec_on p idp
|
||
|
||
|
||
-- Transporting in particular fibrations
|
||
-- -------------------------------------
|
||
|
||
/-
|
||
From the Coq HoTT library:
|
||
|
||
One frequently needs lemmas showing that transport in a certain dependent type is equal to some
|
||
more explicitly defined operation, defined according to the structure of that dependent type.
|
||
For most dependent types, we prove these lemmas in the appropriate file in the types/
|
||
subdirectory. Here we consider only the most basic cases.
|
||
-/
|
||
|
||
-- Transporting in a constant fibration.
|
||
definition transport_const {A B : Type} {x1 x2 : A} (p : x1 ≈ x2) (y : B) :
|
||
transport (λx, B) p y ≈ y :=
|
||
rec_on p idp
|
||
|
||
definition transport2_const {A B : Type} {x1 x2 : A} {p q : x1 ≈ x2} (r : p ≈ q) (y : B) :
|
||
transport_const p y ≈ transport2 (λu, B) r y ⬝ transport_const q y :=
|
||
rec_on r (concat_1p _)^
|
||
|
||
-- Transporting in a pulled back fibration.
|
||
definition transport_compose {A B} {x y : A} (P : B → Type) (f : A → B) (p : x ≈ y) (z : P (f x)) :
|
||
transport (λx, P (f x)) p z ≈ transport P (ap f p) z :=
|
||
rec_on p idp
|
||
|
||
definition transport_precompose {A B C} (f : A → B) (g g' : B → C) (p : g ≈ g') :
|
||
transport (λh : B → C, g ∘ f ≈ h ∘ f) p idp ≈ ap (λh, h ∘ f) p :=
|
||
rec_on p idp
|
||
|
||
definition apD10_ap_precompose {A B C} (f : A → B) (g g' : B → C) (p : g ≈ g') (a : A) :
|
||
apD10 (ap (λh : B → C, h ∘ f) p) a ≈ apD10 p (f a) :=
|
||
rec_on p idp
|
||
|
||
definition apD10_ap_postcompose {A B C} (f : B → C) (g g' : A → B) (p : g ≈ g') (a : A) :
|
||
apD10 (ap (λh : A → B, f ∘ h) p) a ≈ ap f (apD10 p a) :=
|
||
rec_on p idp
|
||
|
||
-- TODO: another example where a term has to be given explicitly
|
||
-- A special case of [transport_compose] which seems to come up a lot.
|
||
definition transport_idmap_ap A (P : A → Type) x y (p : x ≈ y) (u : P x) :
|
||
transport P p u ≈ transport (λz, z) (ap P p) u :=
|
||
rec_on p (idpath (transport (λ (z : Type), z) (ap P (idpath x)) u))
|
||
|
||
|
||
-- The behavior of [ap] and [apD]
|
||
-- ------------------------------
|
||
|
||
-- In a constant fibration, [apD] reduces to [ap], modulo [transport_const].
|
||
definition apD_const {A B} {x y : A} (f : A → B) (p: x ≈ y) :
|
||
apD f p ≈ transport_const p (f x) ⬝ ap f p :=
|
||
rec_on p idp
|
||
|
||
|
||
-- The 2-dimensional groupoid structure
|
||
-- ------------------------------------
|
||
|
||
-- Horizontal composition of 2-dimensional paths.
|
||
definition concat2 {A} {x y z : A} {p p' : x ≈ y} {q q' : y ≈ z} (h : p ≈ p') (h' : q ≈ q') :
|
||
p ⬝ q ≈ p' ⬝ q' :=
|
||
rec_on h (rec_on h' idp)
|
||
|
||
infixl `⬝⬝`:75 := concat2
|
||
|
||
-- 2-dimensional path inversion
|
||
definition inverse2 {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) : p^ ≈ q^ :=
|
||
rec_on h idp
|
||
|
||
-- Whiskering
|
||
-- ----------
|
||
|
||
definition whiskerL {A : Type} {x y z : A} (p : x ≈ y) {q r : y ≈ z} (h : q ≈ r) : p ⬝ q ≈ p ⬝ r :=
|
||
idp ⬝⬝ h
|
||
|
||
definition whiskerR {A : Type} {x y z : A} {p q : x ≈ y} (h : p ≈ q) (r : y ≈ z) : p ⬝ r ≈ q ⬝ r :=
|
||
h ⬝⬝ idp
|
||
|
||
|
||
-- Unwhiskering, a.k.a. cancelling
|
||
-- -------------------------------
|
||
|
||
definition cancelL {A} {x y z : A} (p : x ≈ y) (q r : y ≈ z) : (p ⬝ q ≈ p ⬝ r) → (q ≈ r) :=
|
||
rec_on p (take r, rec_on r (take q a, (concat_1p q)^ ⬝ a)) r q
|
||
|
||
definition cancelR {A} {x y z : A} (p q : x ≈ y) (r : y ≈ z) : (p ⬝ r ≈ q ⬝ r) → (p ≈ q) :=
|
||
rec_on r (take p, rec_on p (take q a, a ⬝ concat_p1 q)) p q
|
||
|
||
-- Whiskering and identity paths.
|
||
|
||
definition whiskerR_p1 {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) :
|
||
(concat_p1 p)^ ⬝ whiskerR h idp ⬝ concat_p1 q ≈ h :=
|
||
rec_on h (rec_on p idp)
|
||
|
||
definition whiskerR_1p {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) :
|
||
whiskerR idp q ≈ idp :> (p ⬝ q ≈ p ⬝ q) :=
|
||
rec_on q idp
|
||
|
||
definition whiskerL_p1 {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) :
|
||
whiskerL p idp ≈ idp :> (p ⬝ q ≈ p ⬝ q) :=
|
||
rec_on q idp
|
||
|
||
definition whiskerL_1p {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) :
|
||
(concat_1p p) ^ ⬝ whiskerL idp h ⬝ concat_1p q ≈ h :=
|
||
rec_on h (rec_on p idp)
|
||
|
||
definition concat2_p1 {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) :
|
||
h ⬝⬝ idp ≈ whiskerR h idp :> (p ⬝ idp ≈ q ⬝ idp) :=
|
||
rec_on h idp
|
||
|
||
definition concat2_1p {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) :
|
||
idp ⬝⬝ h ≈ whiskerL idp h :> (idp ⬝ p ≈ idp ⬝ q) :=
|
||
rec_on h idp
|
||
|
||
-- TODO: note, 4 inductions
|
||
-- The interchange law for concatenation.
|
||
definition concat_concat2 {A : Type} {x y z : A} {p p' p'' : x ≈ y} {q q' q'' : y ≈ z}
|
||
(a : p ≈ p') (b : p' ≈ p'') (c : q ≈ q') (d : q' ≈ q'') :
|
||
(a ⬝⬝ c) ⬝ (b ⬝⬝ d) ≈ (a ⬝ b) ⬝⬝ (c ⬝ d) :=
|
||
rec_on d (rec_on c (rec_on b (rec_on a idp)))
|
||
|
||
definition concat_whisker {A} {x y z : A} (p p' : x ≈ y) (q q' : y ≈ z) (a : p ≈ p') (b : q ≈ q') :
|
||
(whiskerR a q) ⬝ (whiskerL p' b) ≈ (whiskerL p b) ⬝ (whiskerR a q') :=
|
||
rec_on b (rec_on a (concat_1p _)^)
|
||
|
||
-- Structure corresponding to the coherence equations of a bicategory.
|
||
|
||
-- The "pentagonator": the 3-cell witnessing the associativity pentagon.
|
||
definition pentagon {A : Type} {v w x y z : A} (p : v ≈ w) (q : w ≈ x) (r : x ≈ y) (s : y ≈ z) :
|
||
whiskerL p (concat_p_pp q r s)
|
||
⬝ concat_p_pp p (q ⬝ r) s
|
||
⬝ whiskerR (concat_p_pp p q r) s
|
||
≈ concat_p_pp p q (r ⬝ s) ⬝ concat_p_pp (p ⬝ q) r s :=
|
||
rec_on p (take q, rec_on q (take r, rec_on r (take s, rec_on s idp))) q r s
|
||
|
||
-- The 3-cell witnessing the left unit triangle.
|
||
definition triangulator {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) :
|
||
concat_p_pp p idp q ⬝ whiskerR (concat_p1 p) q ≈ whiskerL p (concat_1p q) :=
|
||
rec_on p (take q, rec_on q idp) q
|
||
|
||
definition eckmann_hilton {A : Type} {x:A} (p q : idp ≈ idp :> (x ≈ x)) : p ⬝ q ≈ q ⬝ p :=
|
||
(whiskerR_p1 p ⬝⬝ whiskerL_1p q)^
|
||
⬝ (concat_p1 _ ⬝⬝ concat_p1 _)
|
||
⬝ (concat_1p _ ⬝⬝ concat_1p _)
|
||
⬝ (concat_whisker _ _ _ _ p q)
|
||
⬝ (concat_1p _ ⬝⬝ concat_1p _)^
|
||
⬝ (concat_p1 _ ⬝⬝ concat_p1 _)^
|
||
⬝ (whiskerL_1p q ⬝⬝ whiskerR_p1 p)
|
||
|
||
|
||
-- The action of functions on 2-dimensional paths
|
||
definition ap02 {A B : Type} (f:A → B) {x y : A} {p q : x ≈ y} (r : p ≈ q) : ap f p ≈ ap f q :=
|
||
rec_on r idp
|
||
|
||
definition ap02_pp {A B} (f : A → B) {x y : A} {p p' p'' : x ≈ y} (r : p ≈ p') (r' : p' ≈ p'') :
|
||
ap02 f (r ⬝ r') ≈ ap02 f r ⬝ ap02 f r' :=
|
||
rec_on r (rec_on r' idp)
|
||
|
||
definition ap02_p2p {A B} (f : A→B) {x y z : A} {p p' : x ≈ y} {q q' :y ≈ z} (r : p ≈ p')
|
||
(s : q ≈ q') :
|
||
ap02 f (r ⬝⬝ s) ≈ ap_pp f p q
|
||
⬝ (ap02 f r ⬝⬝ ap02 f s)
|
||
⬝ (ap_pp f p' q')^ :=
|
||
rec_on r (rec_on s (rec_on q (rec_on p idp)))
|
||
|
||
definition apD02 {A : Type} {B : A → Type} {x y : A} {p q : x ≈ y} (f : Π x, B x) (r : p ≈ q) :
|
||
apD f p ≈ transport2 B r (f x) ⬝ apD f q :=
|
||
rec_on r (concat_1p _)^
|
||
end path
|