/-
Copyright (c) 2017 Egbert Rijke. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Egbert Rijke

-/

/-
The goal of this file is to extend the library of pointed types and pointed maps to support the library of prespectra

-/

import types.pointed2 .pointed_pi

open eq pointed

definition psquare_of_phtpy_top {A B C D : Type*}  {ftop : A →* B} {fbot : C →* D} {fleft : A →* C} {fright : B →* D} {ftop' : A →* B} (phtpy : ftop ~* ftop') (psq : psquare ftop' fbot fleft fright) : psquare ftop fbot fleft fright :=
begin
  induction phtpy using phomotopy_rec_on_idp, exact psq,
end

definition psquare_of_phtpy_bot {A B C D : Type*} {ftop : A →* B} {fbot : C →* D} {fleft : A →* C} {fright : B →* D} {fbot' : C →* D} (phtpy : fbot ~* fbot') (psq : psquare ftop fbot' fleft fright) : psquare ftop fbot fleft fright :=
begin
  induction phtpy using phomotopy_rec_on_idp, exact psq,
end

definition psquare_of_phtpy_left {A B C D : Type*} {ftop : A →* B} {fbot : C →* D} {fleft : A →* C} {fright : B →* D} {fleft' : A →* C} (phtpy : fleft ~* fleft') (psq : psquare ftop fbot fleft fright) : psquare ftop fbot fleft' fright :=
begin
  induction phtpy using phomotopy_rec_on_idp, exact psq,
end

definition psquare_of_phtpy_right {A B C D : Type*} {ftop : A →* B} {fbot : C →* D} {fleft : A →* C} {fright : B →* D} {fright' : B →* D} (phtpy : fright ~* fright') (psq : psquare ftop fbot fleft fright) : psquare ftop fbot fleft fright' :=
begin
  induction phtpy using phomotopy_rec_on_idp, exact psq,
end

definition psquare_of_pid_top_bot {A B : Type*} {fleft : A →* B} {fright : A →* B} (phtpy : fright ~* fleft) : psquare (pid A) (pid B) fleft fright :=
psquare_of_phomotopy ((pcompose_pid fright) ⬝* phtpy ⬝* (pid_pcompose fleft)⁻¹*)

--print psquare_of_pid_top_bot

--λ phtpy, psquare_of_phomotopy ((pid_pcompose fleft) ⬝* phtpy ⬝* ((pcompose_pid fright)⁻¹*))

definition psquare_of_pid_left_right {A B : Type*} {ftop : A →* B} {fbot : A →* B} (phtpy : ftop ~* fbot) : psquare ftop fbot (pid A) (pid B) :=
psquare_of_phomotopy ((pid_pcompose ftop) ⬝* phtpy ⬝* ((pcompose_pid fbot)⁻¹*))

--print psquare_of_pid_left_right

definition psquare_hcompose {A B C D E F : Type*} {ftop : A →* B} {fbot : D →* E} {fleft : A →* D} {fright : B →* E} {gtop : B →* C} {gbot : E →* F} {gright : C →* F} (psq_left : psquare ftop fbot fleft fright) (psq_right : psquare gtop gbot fright gright) : psquare (gtop ∘* ftop) (gbot ∘* fbot) fleft gright :=
begin
  fapply psquare_of_phomotopy,
  refine (passoc gright gtop ftop)⁻¹* ⬝* _ ⬝* (passoc gbot fbot fleft)⁻¹*,
  refine (pwhisker_right ftop psq_right) ⬝* (passoc gbot fright ftop) ⬝* _,
  exact (pwhisker_left gbot psq_left),
end

definition psquare_vcompose {A B C D E F : Type*} {ftop : A →* B} {fbot : C →* D} {fleft : A →* C} {fright : B →* D} {gbot : E →* F} {gleft : C →* E} {gright : D →* F} (psq_top : psquare ftop fbot fleft fright) (psq_bot : psquare fbot gbot gleft gright) : psquare ftop gbot (gleft ∘* fleft) (gright ∘* fright) :=
begin
  fapply psquare_of_phomotopy,
  refine (passoc gright fright ftop) ⬝* _ ⬝* (passoc gbot gleft fleft),
  refine (pwhisker_left gright psq_top) ⬝* _,
  refine (passoc gright fbot fleft)⁻¹* ⬝* _,
  exact pwhisker_right fleft psq_bot,
end

definition psquare_of_pconst_top_bot {A B C D : Type*} (fleft : A →* C) (fright : B →* D) : psquare (pconst A B) (pconst C D) fleft fright :=
begin
  fapply psquare_of_phomotopy,
  refine (pcompose_pconst fright) ⬝* _,
  exact (pconst_pcompose fleft)⁻¹*,
end

definition psquare_of_pconst_left_right {A B C D : Type*} (ftop : A →* B) (fbot : C →* D) : psquare ftop fbot (pconst A C) (pconst B D) :=
begin
  fapply psquare_of_phomotopy,
  refine (pconst_pcompose ftop) ⬝* _,
  exact (pcompose_pconst fbot)⁻¹*
end

definition psquare_of_pconst_top_left {A B C D : Type*} (fbot : C →* D) (fright : B →* D) : psquare (pconst A B) fbot (pconst A C) fright :=
begin
  fapply psquare_of_phomotopy,
  refine (pcompose_pconst fright) ⬝* _,
  exact (pcompose_pconst fbot)⁻¹*,
end

definition psquare_of_pconst_bot_right {A B C D : Type*} (ftop : A →* B) (fleft : A →* C) : psquare ftop (pconst C D) fleft (pconst B D) :=
begin
  fapply psquare_of_phomotopy,
  refine (pconst_pcompose ftop) ⬝* _,
  exact (pconst_pcompose fleft)⁻¹*,
end

definition phsquare_of_ppi_homotopy {A B : Type*} {f g h i : A →* B} {phtpy_top : f ~* g} {phtpy_bot : h ~* i} {phtpy_left : f ~* h} {phtpy_right : g ~* i} (H : phtpy_top ⬝* phtpy_right ~~* phtpy_left ⬝* phtpy_bot) : phsquare phtpy_top phtpy_bot phtpy_left phtpy_right :=
  eq_of_ppi_homotopy H

definition ptube_v {A B C D : Type*} {ftop ftop' : A →* B} (phtpy_top : ftop ~* ftop') {fbot fbot' : C →* D} (phtpy_bot : fbot ~* fbot') {fleft : A →* C} {fright : B →* D} (psq_back : psquare ftop fbot fleft fright) (psq_front : psquare ftop' fbot' fleft fright) : Type :=
phsquare (pwhisker_left fright phtpy_top) (pwhisker_right fleft phtpy_bot) psq_back psq_front

definition ptube_h {A B C D : Type*} {ftop : A →* B} {fbot : C →* D} {fleft fleft' : A →* C} (phtpy_left : fleft ~* fleft') {fright fright' : B →* D} (phtpy_right : fright ~* fright') (psq_back : psquare ftop fbot fleft fright) (psq_front : psquare ftop fbot fleft' fright') : Type :=
phsquare (pwhisker_right ftop phtpy_right) (pwhisker_left fbot phtpy_left) psq_back psq_front

--print pinv_right_phomotopy_of_phomotopy

definition psquare_inv_top_bot {A B C D : Type*} {ftop : A ≃* B} {fbot : C ≃* D} {fleft : A →* C} {fright : B →* D} (psq : psquare ftop fbot fleft fright) : psquare ftop⁻¹ᵉ* fbot⁻¹ᵉ* fright fleft :=
begin
  fapply psquare_of_phomotopy,
  refine (pinv_right_phomotopy_of_phomotopy _),
  refine _ ⬝* (passoc fbot⁻¹ᵉ* fright ftop)⁻¹*,
  refine (pinv_left_phomotopy_of_phomotopy _)⁻¹*,
  exact psq,
end

definition p2homotopy_ty_respect_pt {A B : Type*} {f g : A →* B} {H K : f ~* g} (htpy : H ~ K) : Type :=
  begin
    induction H with H p, exact p
  end  = whisker_right (respect_pt g) (htpy pt) ⬝
  begin
induction K with K q, exact q
  end

--print p2homotopy_ty_respect_pt

structure p2homotopy {A B : Type*} {f g : A →* B} (H K : f ~* g) : Type :=
( to_2htpy : H ~ K)
( respect_pt  : p2homotopy_ty_respect_pt to_2htpy)

definition ptube_v_phtpy_bot {A B C D : Type*}
  {ftop ftop' : A →* B} {phtpy_top : ftop ~* ftop'}
  {fbot fbot' : C →* D} {phtpy_bot phtpy_bot' : fbot ~* fbot'} (ppi_htpy_bot : phtpy_bot ~~* phtpy_bot')
  {fleft : A →* C} {fright : B →* D}
  {psq_back : psquare ftop fbot fleft fright}
  {psq_front : psquare ftop' fbot' fleft fright}
  (ptb : ptube_v phtpy_top phtpy_bot psq_back psq_front)
  : ptube_v phtpy_top phtpy_bot' psq_back psq_front
  :=
begin
  induction ppi_htpy_bot using ppi_homotopy_rec_on_idp,
  exact ptb,
end

definition ptube_v_eq_bot {A B C D : Type*} {ftop ftop' : A →* B} (htpy_top : ftop ~* ftop') {fbot fbot' : C →* D} {htpy_bot htpy_bot' : fbot ~* fbot'} (p : htpy_bot = htpy_bot') {fleft : A →* C} {fright : B →* D} (psq_back : psquare ftop fbot fleft fright) (psq_front : psquare ftop' fbot' fleft fright) :
  ptube_v htpy_top htpy_bot psq_back psq_front → ptube_v htpy_top htpy_bot' psq_back psq_front :=
begin
  induction p,
  exact id,
end

definition ptube_v_left_inv {A B C D : Type*} {ftop : A ≃* B} {fbot : C ≃* D} {fleft : A →* C} {fright : B →* D}
  (psq : psquare ftop fbot fleft fright) :
  ptube_v
    (pleft_inv ftop)
    (pleft_inv fbot)
    (psquare_hcompose psq (psquare_inv_top_bot psq))
    (psquare_of_pid_top_bot phomotopy.rfl) :=
begin
  refine ptube_v_phtpy_bot _ _,
  exact pleft_inv fbot,
  exact ppi_homotopy.rfl,
  fapply phsquare_of_ppi_homotopy, repeat exact sorry,
end