Require Import Basics.
[Loading ML file number_string_notation_plugin.cmxs (using legacy method) ... done]
Require Import Types.
Require Import WildCat.
Require Import Pointed.Core.

Local Open Scope pointed_scope.

(* Pointed equivalence is a reflexive relation. *)
Global Instance pequiv_reflexive : Reflexive pEquiv.
Reflexive pEquiv
Proof.
Reflexive pEquiv
  intro; apply pequiv_pmap_idmap.
Defined.

(* We can probably get rid of the following notation, and use ^-1$ instead. *)
Notation "f ^-1*" := (@cate_inv pType _ _ _ _ hasequivs_ptype _ _ f) : pointed_scope.

(* Pointed equivalence is a symmetric relation. *)
Global Instance pequiv_symmetric : Symmetric pEquiv.
Symmetric pEquiv
Proof.
Symmetric pEquiv
  intros ? ?; apply pequiv_inverse.
Defined.

(* Pointed equivalences compose. *)
Definition pequiv_compose {A B C : pType} (f : A <~>* B) (g : B <~>* C)
  : A <~>* C
  := g $oE f.

(* Pointed equivalence is a transitive relation. *)
Global Instance pequiv_transitive : Transitive pEquiv.
Transitive pEquiv
Proof.
Transitive pEquiv
  intros ? ? ?; apply pequiv_compose.
Defined.

Notation "g o*E f" := (pequiv_compose f g) : pointed_scope.

(* Sometimes we wish to construct a pEquiv from an equiv and a proof that it is pointed. *)
Definition Build_pEquiv' {A B : pType} (f : A <~> B)
  (p : f (point A) = point B)
  : A <~>* B := Build_pEquiv _ _ (Build_pMap _ _ f p) _.

Arguments Build_pEquiv' & _ _ _ _.

(* A version of equiv_adjointify for pointed equivalences where all data is pointed. There is a lot of unnecessary data here but sometimes it is easier to prove equivalences using this. *)
Definition pequiv_adjointify {A B : pType} (f : A ->* B) (f' : B ->* A)
  (r : f o* f' ==* pmap_idmap) (s : f' o* f == pmap_idmap) : A <~>* B
  := (Build_pEquiv _ _ f (isequiv_adjointify f f' r s)).

(* In some situations you want the back and forth maps to be pointed but not the sections. *)
Definition pequiv_adjointify' {A B : pType} (f : A ->* B) (f' : B ->* A)
  (r : f o f' == idmap) (s : f' o f == idmap) : A <~>* B
  := (Build_pEquiv _ _ f (isequiv_adjointify f f' r s)).

(** Pointed versions of [moveR_equiv_M] and friends. *)
Definition moveR_pequiv_Mf {A B C} (f : B <~>* C) (g : A ->* B) (h : A ->* C)
           (p : g ==* f^-1* o* h)
  : (f o* g ==* h).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: g ==* f^-1* o* h
f o* g ==* h
Proof.
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: g ==* f^-1* o* h
f o* g ==* h
  refine (pmap_postwhisker f p @* _).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: g ==* f^-1* o* h
f o* (f^-1* o* h) ==* h
  refine ((pmap_compose_assoc _ _ _)^* @* _).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: g ==* f^-1* o* h
f o* f^-1* o* h ==* h
  refine (pmap_prewhisker h (peisretr f) @* _).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: g ==* f^-1* o* h
pmap_idmap o* h ==* h
  apply pmap_postcompose_idmap.
Defined.

Definition moveL_pequiv_Mf {A B C} (f : B <~>* C) (g : A ->* B) (h : A ->* C)
           (p : f^-1* o* h ==* g)
  : (h ==* f o* g).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: f^-1* o* h ==* g
h ==* f o* g
Proof.
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: f^-1* o* h ==* g
h ==* f o* g
  refine (_ @* pmap_postwhisker f p).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: f^-1* o* h ==* g
h ==* f o* (f^-1* o* h)
  refine (_ @* (pmap_compose_assoc _ _ _)).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: f^-1* o* h ==* g
h ==* f o* f^-1* o* h
  refine ((pmap_postcompose_idmap _)^* @* _).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: f^-1* o* h ==* g
pmap_idmap o* h ==* f o* f^-1* o* h
  apply pmap_prewhisker.
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: f^-1* o* h ==* g
pmap_idmap ==* f o* f^-1*
  symmetry; apply peisretr.
Defined.

Definition moveL_pequiv_Vf {A B C} (f : B <~>* C) (g : A ->* B) (h : A ->* C)
           (p : f o* g ==* h)
  : g ==* f^-1* o* h.
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: f o* g ==* h
g ==* f^-1* o* h
Proof.
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: f o* g ==* h
g ==* f^-1* o* h
  refine (_ @* pmap_postwhisker f^-1* p).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: f o* g ==* h
g ==* f^-1* o* (f o* g)
  refine (_ @* (pmap_compose_assoc _ _ _)).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: f o* g ==* h
g ==* f^-1* o* f o* g
  refine ((pmap_postcompose_idmap _)^* @* _).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: f o* g ==* h
pmap_idmap o* g ==* f^-1* o* f o* g
  apply pmap_prewhisker.
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: f o* g ==* h
pmap_idmap ==* f^-1* o* f
  symmetry; apply peissect.
Defined.

Definition moveR_pequiv_Vf {A B C} (f : B <~>* C) (g : A ->* B) (h : A ->* C)
           (p : h ==* f o* g)
   : f^-1* o* h ==* g.
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: h ==* f o* g
f^-1* o* h ==* g
Proof.
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: h ==* f o* g
f^-1* o* h ==* g
  refine (pmap_postwhisker f^-1* p @* _).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: h ==* f o* g
f^-1* o* (f o* g) ==* g
  refine ((pmap_compose_assoc _ _ _)^* @* _).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: h ==* f o* g
f^-1* o* f o* g ==* g
  refine (pmap_prewhisker g (peissect f) @* _).
A, B, C: pType
f: B <~>* C
g: A ->* B
h: A ->* C
p: h ==* f o* g
pmap_idmap o* g ==* g
  apply pmap_postcompose_idmap.
Defined.

Definition moveR_pequiv_fV {A B C} (f : B ->* C) (g : A <~>* B) (h : A ->* C)
           (p : f o* g ==* h)
  : (f ==* h o* g^-1*).
A, B, C: pType
f: B ->* C
g: A <~>* B
h: A ->* C
p: f o* g ==* h
f ==* h o* g^-1*
Proof.
A, B, C: pType
f: B ->* C
g: A <~>* B
h: A ->* C
p: f o* g ==* h
f ==* h o* g^-1*
  refine (_ @* pmap_prewhisker g^-1* p).
A, B, C: pType
f: B ->* C
g: A <~>* B
h: A ->* C
p: f o* g ==* h
f ==* f o* g o* g^-1*
  refine (_ @* (pmap_compose_assoc _ _ _)^*).
A, B, C: pType
f: B ->* C
g: A <~>* B
h: A ->* C
p: f o* g ==* h
f ==* f o* (g o* g^-1*)
  refine ((pmap_precompose_idmap _)^* @* _).
A, B, C: pType
f: B ->* C
g: A <~>* B
h: A ->* C
p: f o* g ==* h
f o* pmap_idmap ==* f o* (g o* g^-1*)
  apply pmap_postwhisker.
A, B, C: pType
f: B ->* C
g: A <~>* B
h: A ->* C
p: f o* g ==* h
pmap_idmap ==* g o* g^-1*
  symmetry; apply peisretr.
Defined.

Definition pequiv_pequiv_precompose `{Funext} {A B C : pType} (f : A <~>* B)
  : (B ->** C) <~>* (A ->** C).
H: Funext
A, B, C: pType
f: A <~>* B
(B ->** C) <~>* (A ->** C)
Proof.
H: Funext
A, B, C: pType
f: A <~>* B
(B ->** C) <~>* (A ->** C)
  srapply Build_pEquiv'.
H: Funext
A, B, C: pType
f: A <~>* B
(B ->** C) <~> (A ->** C)
H: Funext
A, B, C: pType
f: A <~>* B
?f pt = pt
  -
H: Funext
A, B, C: pType
f: A <~>* B
(B ->** C) <~> (A ->** C) exact (equiv_precompose_cat_equiv f).
  - (* By using [pelim f], we can avoid [Funext] in this part of the proof. *)
H: Funext
A, B, C: pType
f: A <~>* B
equiv_precompose_cat_equiv f pt = pt
    cbn; unfold "o*", point_pforall; cbn.
H: Funext
A, B, C: pType
f: A <~>* B
Build_pMap A C (fun _ : A => pt)
  (ap (fun _ : B => pt) (point_eq f) @ 1) = pconst
    by pelim f.
Defined.

Definition pequiv_pequiv_postcompose `{Funext} {A B C : pType} (f : B <~>* C)
  : (A ->** B) <~>* (A ->** C).
H: Funext
A, B, C: pType
f: B <~>* C
(A ->** B) <~>* (A ->** C)
Proof.
H: Funext
A, B, C: pType
f: B <~>* C
(A ->** B) <~>* (A ->** C)
  srapply Build_pEquiv'.
H: Funext
A, B, C: pType
f: B <~>* C
(A ->** B) <~> (A ->** C)
H: Funext
A, B, C: pType
f: B <~>* C
?f pt = pt
  -
H: Funext
A, B, C: pType
f: B <~>* C
(A ->** B) <~> (A ->** C) exact (equiv_postcompose_cat_equiv f).
  -
H: Funext
A, B, C: pType
f: B <~>* C
equiv_postcompose_cat_equiv f pt = pt cbn; unfold "o*", point_pforall; cbn.
H: Funext
A, B, C: pType
f: B <~>* C
Build_pMap A C (fun _ : A => f pt) (1 @ point_eq f) =
pconst
    by pelim f.
Defined.

Proposition equiv_pequiv_inverse `{Funext} {A B : pType}
  : (A <~>* B) <~> (B <~>* A).
H: Funext
A, B: pType
(A <~>* B) <~> (B <~>* A)
Proof.
H: Funext
A, B: pType
(A <~>* B) <~> (B <~>* A)
  refine (issig_pequiv' _ _ oE _ oE (issig_pequiv' A B)^-1).
H: Funext
A, B: pType
{f : A <~> B & f pt = pt} <~>
{f : B <~> A & f pt = pt}
  srapply (equiv_functor_sigma' (equiv_equiv_inverse _ _)); intro e; cbn.
H: Funext
A, B: pType
e: A <~> B
e pt = pt <~> e^-1 pt = pt
  exact (equiv_moveR_equiv_V _ _ oE equiv_path_inverse _ _).
Defined.