lean2/library/hott/path_groupoids.lean

-- 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

import .logic .tactic

using tactic

-- Path
-- ----

inductive path {A : Type} (a : A) : A → Type :=
| idpath : path a a

infix `≈`:50 := path
notation x `≈` y:50 `:>`:0 A:0 := @path A x y    -- TODO: is this right?
notation `idp`:max := idpath _    -- TODO: can we / should we use `1`?

namespace path
  abbreviation induction_on {A : Type} {a b : A} (p : a ≈ b)
    {C : Π (b : A) (p : a ≈ b), Type} (H : C a (idpath a)) : C b p :=
  path_rec H p
end path


-- TODO: should all this be in namespace path?
using 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

-- TODO: should this be an abbreviation?
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 :=
induction_on p idp

-- The identity path is a right unit. 
definition concat_1p {A : Type} {x y : A} (p : x ≈ y) : idp @ p ≈ p :=
induction_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 :=
induction_on r (induction_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) :=
induction_on r (induction_on q idp)

-- The left inverse law.
definition concat_pV {A : Type} {x y : A} (p : x ≈ y) : p @ p^ ≈ idp :=
induction_on p idp

-- The right inverse law.
definition concat_Vp {A : Type} {x y : A} (p : x ≈ y) : p^ @ p ≈ idp :=
induction_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 :=
induction_on q (induction_on p idp)

definition concat_p_Vp {A : Type} {x y z : A} (p : x ≈ y) (q : x ≈ z) : p @ (p^ @ q) ≈ q :=
induction_on q (induction_on p idp)

definition concat_pp_V {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) : (p @ q) @ q^ ≈ p :=
induction_on q (induction_on p idp)

definition concat_pV_p {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) : (p @ q^) @ q ≈ p :=
induction_on q (take p, induction_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^ :=
induction_on q (induction_on p idp)

definition inv_Vp {A : Type} {x y z : A} (p : y ≈ x) (q : y ≈ z) : (p^ @ q)^ ≈ q^ @ p :=
induction_on q (induction_on p idp)

-- universe metavariables
definition inv_pV {A : Type} {x y z : A} (p : x ≈ y) (q : z ≈ y) : (p @ q^)^ ≈ q @ p^ :=
induction_on p (λq, induction_on q idp) q

definition inv_VV {A : Type} {x y z : A} (p : y ≈ x) (q : z ≈ y) : (p^ @ q^)^ ≈ q @ p :=
induction_on p (induction_on q idp)

-- Inverse is an involution.
definition inv_V {A : Type} {x y : A} (p : x ≈ y) : p^^ ≈ p :=
induction_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
  induction_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 :=
induction_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 :=
induction_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 :=
induction_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 := 
induction_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 :=
induction_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 :=
induction_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^ :=
induction_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 :=
induction_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 :=
induction_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^ :=
induction_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^ :=
induction_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 :=
induction_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 :=
induction_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 :=
induction_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 :=
induction_on p (take q h, h @ (concat_1p _)) q


-- Transport
-- ---------

-- keep transparent, so transport _ idp p is definitionally equal to p
abbreviation transport {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (u : P x) : P y :=
path.induction_on p u

definition transport_1 {A : Type} (P : A → Type) {x : A} (u : P x) : transport _ idp u ≈ u :=
idp

-- 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.induction_on p idp

-- TODO: is this better than an alias? Note use of curly brackets
abbreviation ap01 := ap

abbreviation 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.induction_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 :=
induction_on H (induction_on p idp)

definition apD {A:Type} {B : A → Type} (f : Πa:A, B a) {x y : A} (p : x ≈ y) : p # (f x) ≈ f y :=
induction_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 :=
induction_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 :=
induction_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 :=
induction_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 :=
induction_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) :=
induction_on q (induction_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) :=
induction_on p (take r q, induction_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) :=
induction_on p (take q, induction_on q (take r, concat_pp_p _ _ _)) q r
-- induction_on p (take q : x ≈ z, induction_on q (take r : f x ≈ f w, concat_pp_p idp idp r)) q r

-- TODO: can we do this proof using tactics?
-- proof
--   apply (induction_on p _ q r),
--   apply (take q, induction_on q _),
--   apply (concat_pp_p idp idp r)
-- qed

-- proof
--   apply (induction_on p _ q r),
--   take q, proof 
--     apply (induction_on q),
--     take r, 
--     apply concat_pp_p
--   qed  
-- qed

-- Coq proof:
-- Proof.
--   destruct p, q. simpl. exact (concat_pp_p idp idp r).
-- Defined.

-- 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^) :=
induction_on p idp

definition ap_V {A B : Type} (f : A → B) {x y : A} (p : x ≈ y) : ap f (p^) ≈ (ap f p)^ :=
induction_on p idp

-- [ap] itself is functorial in the first argument.

-- TODO: rename id to idmap?
definition ap_idmap {A : Type} {x y : A} (p : x ≈ y) : ap id p ≈ p :=
induction_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) :=
induction_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) := 
induction_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 :=
induction_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) :=
induction_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 :=
induction_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) :=
induction_on q (concat_p1 _ @ (concat_1p _)^)

-- TODO: move these
calc_subst transport
calc_trans concat
calc_refl idpath

-- 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) :=
induction_on q (induction_on s
  (calc 
    (r @ (ap f q)) @ ((p y) @ (idpath (g y))) ≈ (r @ (ap f q)) @ (p y) : {concat_p1 _}
      ... ≈ r @ ((ap f q) @ (p y)) : concat_pp_p _ _ _
      ... ≈ r @ ((p x) @ (ap g q)) : {concat_Ap p q}
      ... ≈ (r @ (p x)) @ ((ap g q)) : concat_p_pp _ _ _
      ... ≈ (r @ (p x)) @ ((ap g q) @ (idpath (g y))) : {concat_p1 _}))

-- TODO: the Coq proof of the previous theorem is much shorter:
-- Proof.
--   destruct q, s; simpl.
--   repeat rewrite concat_p1.
--   reflexivity.
-- Defined.

(-- TODO: These are similar

definition concat_pA_p {A B : Type} {f g : A → B} (p : forall x, f x ≈ g x)
  {x y : A} (q : x ≈ y)
  {w : B} (r : w ≈ f x)
  :
  (r @ ap f q) @ p y ≈ (r @ p x) @ ap g q.
Proof.
  destruct q; simpl.
  repeat rewrite concat_p1.
  reflexivity.
Defined.

definition concat_A_pp {A B : Type} {f g : A → B} (p : forall x, f x ≈ g x)
  {x y : A} (q : x ≈ y)
  {z : B} (s : g y ≈ z)
  :
  (ap f q) @ (p y @ s) ≈ (p x) @ (ap g q @ s).
Proof.
  destruct q, s; simpl.
  repeat rewrite concat_p1, concat_1p.
  reflexivity.
Defined.

definition concat_pA1_pp {A : Type} {f : A → A} (p : forall x, f x ≈ x)
  {x y : A} (q : x ≈ y)
  {w z : A} (r : w ≈ f x) (s : y ≈ z)
  :
  (r @ ap f q) @ (p y @ s) ≈ (r @ p x) @ (q @ s).
Proof.
  destruct q, s; simpl.
  repeat rewrite concat_p1.
  reflexivity.
Defined.

definition concat_pp_A1p {A : Type} {g : A → A} (p : forall x, x ≈ g x)
  {x y : A} (q : x ≈ y)
  {w z : A} (r : w ≈ x) (s : g y ≈ z)
  :
  (r @ p x) @ (ap g q @ s) ≈ (r @ q) @ (p y @ s).
Proof.
  destruct q, s; simpl.
  repeat rewrite concat_p1.
  reflexivity.
Defined.

definition concat_pA1_p {A : Type} {f : A → A} (p : forall x, f x ≈ x)
  {x y : A} (q : x ≈ y)
  {w : A} (r : w ≈ f x)
  :
  (r @ ap f q) @ p y ≈ (r @ p x) @ q.
Proof.
  destruct q; simpl.
  repeat rewrite concat_p1.
  reflexivity.
Defined.

definition concat_A1_pp {A : Type} {f : A → A} (p : forall x, f x ≈ x)
  {x y : A} (q : x ≈ y)
  {z : A} (s : y ≈ z)
  :
  (ap f q) @ (p y @ s) ≈ (p x) @ (q @ s).
Proof.
  destruct q, s; simpl.
  repeat rewrite concat_p1, concat_1p.
  reflexivity.
Defined.

definition concat_pp_A1 {A : Type} {g : A → A} (p : forall x, x ≈ g x)
  {x y : A} (q : x ≈ y)
  {w : A} (r : w ≈ x)
  :
  (r @ p x) @ ap g q ≈ (r @ q) @ p y.
Proof.
  destruct q; simpl.
  repeat rewrite concat_p1.
  reflexivity.
Defined.

definition concat_p_A1p {A : Type} {g : A → A} (p : forall x, x ≈ g x)
  {x y : A} (q : x ≈ y)
  {z : A} (s : g y ≈ z)
  :
  p x @ (ap g q @ s) ≈ q @ (p y @ s).
Proof.
  destruct q, s; simpl.
  repeat rewrite concat_p1, concat_1p.
  reflexivity.
Defined.
--)


-- 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 :=
induction_on h (take h', induction_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)^ := 
induction_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:=
induction_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 :=
induction_on q (induction_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)


-----------------------------------------------
-- TODO: 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 := 
induction_on p (take z q, induction_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 := 
induction_on p (take y q, induction_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 := 
induction_on p (take y q, induction_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 := 
induction_on p (take z q, induction_on q (take w r, induction_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

-- old proof:
-- induction_on q (take p, induction_on p idp) p

-- This is an attempt to do it as a tactic proof
-- proof
--   apply (induction_on q _ p),
--   take p,
--   apply (induction_on p idp)
-- qed

-- set_option unifier.max_steps 1000000

-- TODO: update: this no longer works, but it used to (it took a couple of seconds)

-- theorem test {A : Type} (P : A → Type) (x : A) (u_1 : P x) :
-- path
--     (concat
--        (concat (ap (λ (e : path x x), transport P e u_1) (concat_p_pp (idpath x) (idpath x) (idpath x)))
--           (transport_pp P (concat (idpath x) (idpath x)) (idpath x) u_1))
--        (ap (transport P (idpath x)) (transport_pp P (idpath x) (idpath x) u_1)))
--     (concat (transport_pp P (idpath x) (concat (idpath x) (idpath x)) u_1)
--        (transport_pp P (idpath x) (idpath x) (transport P (idpath x) u_1))) := 
-- 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 _ u
-- triple_induction p q r (take u, test P x u) u

-- this doesn't work, even with max_steps = 100000
--double_induction p q (take r, induction_on r (take u, test P x u))) r u

-- even this doesn't work!
-- 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) :=
-- have aux:
--  Πu_1 : P x, path
--     (concat
--        (concat (ap (λ (e : path x x), transport' e u_1) (concat_p_pp (idpath x) (idpath x) (idpath x)))
--           (transport_pp P (concat (idpath x) (idpath x)) (idpath x) u_1))
--        (ap (transport P (idpath x)) (transport_pp P (idpath x) (idpath x) u_1)))
--     (concat (transport_pp P (idpath x) (concat (idpath x) (idpath x)) u_1)
--        (transport_pp P (idpath x) (idpath x) (transport' (idpath x) u_1))), from take u_1, idp,
-- triple_induction p q r (take u, aux u) u

-- This works with max_steps = 10000
-- 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) :=
-- induction_on p (take q, induction_on q (take r, induction_on r (take u, test P x u))) q r u

-- This shows the goal, with max_steps = 100000
-- 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) :=
-- induction_on p (take q, induction_on q (take r, induction_on r (take u, _)))) q r u



-- Back to the regular development

-- In the future, we may expect to need some higher coherence for transport:
--  for instance, that transport acting on the associator is trivial.

-- set_option unifier.max_steps 1000000

-- TODO: we cannot get this yet
-- in Coq, the type annotation is only for clarity -- it is not needed
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) := 
sorry

(-- Coq proof:
Proof.
  destruct p, q, r.  simpl.  exact 1.
Defined.
--)

--  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)
:= induction_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) := 
induction_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 :=
induction_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 :=
induction_on r1 (induction_on r2 idp)

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)^) :=
induction_on r idp

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 :=
induction_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)) :=
induction_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 :=
induction_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 :=
induction_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 :=
induction_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 :=
induction_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) :=
induction_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) :=
induction_on p idp

-- 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 :=
induction_on p idp


-- 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 :=
induction_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' :=
induction_on h (induction_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^ :=
induction_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) :=
induction_on p (take r, induction_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) :=
induction_on r (take p, induction_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 :=
induction_on h (induction_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) :=
induction_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) :=
induction_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 :=
induction_on h (induction_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) :=
induction_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) :=
induction_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) :=
induction_on d (induction_on c (induction_on b (induction_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') :=
induction_on b (induction_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 :=
sorry
-- induction_on p (induction_on q (induction_on r (induction_on s idp)))

(-- The Coq proof:

-- 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.
Proof.
  case p, q, r, s.  reflexivity.
Defined.

--)

-- 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) :=
induction_on p (take q, induction_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 :=
induction_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' := 
induction_on r (induction_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')^ :=
sorry
-- induction_on r (induction_on s (induction_on p (induction_on q 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 :=
induction_on r (concat_1p _)^

-- And now for a lemma whose statement is much longer than its proof.
definition apD02_pp {A} (B : A → Type) (f : Π x:A, B x) {x y : A}
    {p1 p2 p3 : x ≈ y} (r1 : p1 ≈ p2) (r2 : p2 ≈ p3) : 
  apD02 f (r1 @ r2) ≈ apD02 f r1
    @ whiskerL (transport2 B r1 (f x)) (apD02 f r2)
    @ concat_p_pp _ _ _
    @ (whiskerR ((transport2_p2p B r1 r2 (f x))^) (apD f p3)) :=
sorry
-- induction_on r1 (take r2, induction_on r2 (take p1, induction_on p1 idp)) r2 p1

(--

The Coq proof:
definition apD02_pp {A} (B : A → Type) (f : Π x:A, B x) {x y : A}
  {p1 p2 p3 : x ≈ y} (r1 : p1 ≈ p2) (r2 : p2 ≈ p3)
  : apD02 f (r1 @ r2)
  ≈ apD02 f r1
  @ whiskerL (transport2 B r1 (f x)) (apD02 f r2)
  @ concat_p_pp _ _ _
  @ (whiskerR (transport2_p2p B r1 r2 (f x))^ (apD f p3)).
Proof.
  destruct r1, r2. destruct p1. reflexivity.
Defined.

--)

(--
-- ** Tactics, hints, and aliases

-- [concat], with arguments flipped. Useful mainly in the idiom [apply (concatR (expression))]. Given as a notation not a definition so that the resultant terms are literally instances of [concat], with no unfolding required.
Notation concatR := (λp q, concat q p).

Hint Resolve
  concat_1p concat_p1 concat_p_pp
  inv_pp inv_V
 : path_hints.

(* First try at a paths db
We want the RHS of the equation to become strictly simpler
Hint Rewrite
@concat_p1
@concat_1p
@concat_p_pp (* there is a choice here !*)
@concat_pV
@concat_Vp
@concat_V_pp
@concat_p_Vp
@concat_pp_V
@concat_pV_p
(*@inv_pp*) (* I am not sure about this one
@inv_V
@moveR_Mp
@moveR_pM
@moveL_Mp
@moveL_pM
@moveL_1M
@moveL_M1
@moveR_M1
@moveR_1M
@ap_1
(* @ap_pp
@ap_p_pp ?*)
@inverse_ap
@ap_idmap
(* @ap_compose
@ap_compose'*)
@ap_const
(* Unsure about naturality of [ap], was absent in the old implementation*)
@apD10_1
:paths.

Ltac hott_simpl :=
  autorewrite with paths in * |- * ; auto with path_hints.

--)