/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Floris van Doorn, Jakob von Raumer

Cubes
-/

import .square

open equiv equiv.ops is_equiv sigma sigma.ops

namespace eq

  inductive cube {A : Type} {a₀₀₀ : A} : Π{a₂₀₀ a₀₂₀ a₂₂₀ a₀₀₂ a₂₀₂ a₀₂₂ a₂₂₂ : A}
       {p₁₀₀ : a₀₀₀ = a₂₀₀} {p₀₁₀ : a₀₀₀ = a₀₂₀} {p₀₀₁ : a₀₀₀ = a₀₀₂}
       {p₁₂₀ : a₀₂₀ = a₂₂₀} {p₂₁₀ : a₂₀₀ = a₂₂₀} {p₂₀₁ : a₂₀₀ = a₂₀₂}
       {p₁₀₂ : a₀₀₂ = a₂₀₂} {p₀₁₂ : a₀₀₂ = a₀₂₂} {p₀₂₁ : a₀₂₀ = a₀₂₂}
       {p₁₂₂ : a₀₂₂ = a₂₂₂} {p₂₁₂ : a₂₀₂ = a₂₂₂} {p₂₂₁ : a₂₂₀ = a₂₂₂}
       (s₀₁₁ : square p₀₁₀ p₀₁₂ p₀₀₁ p₀₂₁)
       (s₂₁₁ : square p₂₁₀ p₂₁₂ p₂₀₁ p₂₂₁)
       (s₁₀₁ : square p₁₀₀ p₁₀₂ p₀₀₁ p₂₀₁)
       (s₁₂₁ : square p₁₂₀ p₁₂₂ p₀₂₁ p₂₂₁)
       (s₁₁₀ : square p₀₁₀ p₂₁₀ p₁₀₀ p₁₂₀)
       (s₁₁₂ : square p₀₁₂ p₂₁₂ p₁₀₂ p₁₂₂), Type :=
  idc : cube ids ids ids ids ids ids

  variables {A B : Type} {a₀₀₀ a₂₀₀ a₀₂₀ a₂₂₀ a₀₀₂ a₂₀₂ a₀₂₂ a₂₂₂ a a' : A}
            {p₁₀₀ : a₀₀₀ = a₂₀₀} {p₀₁₀ : a₀₀₀ = a₀₂₀} {p₀₀₁ : a₀₀₀ = a₀₀₂}
            {p₁₂₀ : a₀₂₀ = a₂₂₀} {p₂₁₀ : a₂₀₀ = a₂₂₀} {p₂₀₁ : a₂₀₀ = a₂₀₂}
            {p₁₀₂ : a₀₀₂ = a₂₀₂} {p₀₁₂ : a₀₀₂ = a₀₂₂} {p₀₂₁ : a₀₂₀ = a₀₂₂}
            {p₁₂₂ : a₀₂₂ = a₂₂₂} {p₂₁₂ : a₂₀₂ = a₂₂₂} {p₂₂₁ : a₂₂₀ = a₂₂₂}
            {s₁₁₀ : square p₀₁₀ p₂₁₀ p₁₀₀ p₁₂₀}
            {s₁₁₂ : square p₀₁₂ p₂₁₂ p₁₀₂ p₁₂₂}
            {s₀₁₁ : square p₀₁₀ p₀₁₂ p₀₀₁ p₀₂₁}
            {s₂₁₁ : square p₂₁₀ p₂₁₂ p₂₀₁ p₂₂₁}
            {s₁₀₁ : square p₁₀₀ p₁₀₂ p₀₀₁ p₂₀₁}
            {s₁₂₁ : square p₁₂₀ p₁₂₂ p₀₂₁ p₂₂₁}
            {b₁ b₂ b₃ b₄ : B}
            (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂)

  definition idc    [reducible] [constructor]         := @cube.idc
  definition idcube [reducible] [constructor] (a : A) := @cube.idc A a

  variables (s₁₁₀ s₁₀₁)
  definition refl1 : cube s₀₁₁ s₀₁₁ hrfl hrfl vrfl vrfl :=
  by induction s₀₁₁; exact idc

  definition refl2 : cube hrfl hrfl s₁₀₁ s₁₀₁ hrfl hrfl :=
  by induction s₁₀₁; exact idc
  
  definition refl3 : cube vrfl vrfl vrfl vrfl s₁₁₀ s₁₁₀ :=
  by induction s₁₁₀; exact idc

  variables {s₁₁₀ s₁₀₁}
  definition rfl1 : cube s₀₁₁ s₀₁₁ hrfl hrfl vrfl vrfl := !refl1

  definition rfl2 : cube hrfl hrfl s₁₀₁ s₁₀₁ hrfl hrfl := !refl2

  definition rfl3 : cube vrfl vrfl vrfl vrfl s₁₁₀ s₁₁₀ := !refl3

  -- Variables for composition
  variables {a₄₀₀ a₄₀₂ a₄₂₀ a₄₂₂ a₀₄₀ a₀₄₂ a₂₄₀ a₂₄₂ a₀₀₄ a₀₂₄ a₂₀₄ a₂₂₄ : A}
    {p₃₀₀ : a₂₀₀ = a₄₀₀} {p₃₀₂ : a₂₀₂ = a₄₀₂} {p₃₂₀ : a₂₂₀ = a₄₂₀} {p₃₂₂ : a₂₂₂ = a₄₂₂}
    {p₄₀₁ : a₄₀₀ = a₄₀₂} {p₄₁₀ : a₄₀₀ = a₄₂₀} {p₄₁₂ : a₄₀₂ = a₄₂₂} {p₄₂₁ : a₄₂₀ = a₄₂₂}
    {p₀₃₀ : a₀₂₀ = a₀₄₀} {p₀₃₂ : a₀₂₂ = a₀₄₂} {p₂₃₀ : a₂₂₀ = a₂₄₀} {p₂₃₂ : a₂₂₂ = a₂₄₂}
    {p₀₄₁ : a₀₄₀ = a₀₄₂} {p₁₄₀ : a₀₄₀ = a₂₄₀} {p₁₄₂ : a₀₄₂ = a₂₄₂} {p₂₄₁ : a₂₄₀ = a₂₄₂}
    {p₀₀₃ : a₀₀₂ = a₀₀₄} {p₀₂₃ : a₀₂₂ = a₀₂₄} {p₂₀₃ : a₂₀₂ = a₂₀₄} {p₂₂₃ : a₂₂₂ = a₂₂₄}
    {p₀₁₄ : a₀₀₄ = a₀₂₄} {p₁₀₄ : a₀₀₄ = a₂₀₄} {p₁₂₄ : a₀₂₄ = a₂₂₄} {p₂₁₄ : a₂₀₄ = a₂₂₄}
    {s₃₀₁ : square p₃₀₀ p₃₀₂ p₂₀₁ p₄₀₁} {s₃₁₀ : square p₂₁₀ p₄₁₀ p₃₀₀ p₃₂₀}
    {s₃₁₂ : square p₂₁₂ p₄₁₂ p₃₀₂ p₃₂₂} {s₃₂₁ : square p₃₂₀ p₃₂₂ p₂₂₁ p₄₂₁}
    {s₄₁₁ : square p₄₁₀ p₄₁₂ p₄₀₁ p₄₂₁}
    {s₀₃₁ : square p₀₃₀ p₀₃₂ p₀₂₁ p₀₄₁} {s₁₃₀ : square p₀₃₀ p₂₃₀ p₁₂₀ p₁₄₀}
    {s₁₃₂ : square p₀₃₂ p₂₃₂ p₁₂₂ p₁₄₂} {s₂₃₁ : square p₂₃₀ p₂₃₂ p₂₂₁ p₂₄₁}
    {s₁₄₁ : square p₁₄₀ p₁₄₂ p₀₄₁ p₂₄₁}
    {s₀₁₃ : square p₀₁₂ p₀₁₄ p₀₀₃ p₀₂₃} {s₁₀₃ : square p₁₀₂ p₁₀₄ p₀₀₃ p₂₀₃}
    {s₁₂₃ : square p₁₂₂ p₁₂₄ p₀₂₃ p₂₂₃} {s₂₁₃ : square p₂₁₂ p₂₁₄ p₂₀₃ p₂₂₃}
    {s₁₁₄ : square p₀₁₄ p₂₁₄ p₁₀₄ p₁₂₄}
    (d : cube s₂₁₁ s₄₁₁ s₃₀₁ s₃₂₁ s₃₁₀ s₃₁₂)
    (e : cube s₀₃₁ s₂₃₁ s₁₂₁ s₁₄₁ s₁₃₀ s₁₃₂)
    (f : cube s₀₁₃ s₂₁₃ s₁₀₃ s₁₂₃ s₁₁₂ s₁₁₄)

  /- Composition of Cubes -/

  include c d
  definition cube_concat1 : cube s₀₁₁ s₄₁₁ (s₁₀₁ ⬝h s₃₀₁) (s₁₂₁ ⬝h s₃₂₁) (s₁₁₀ ⬝v s₃₁₀) (s₁₁₂ ⬝v s₃₁₂) :=
  by induction d; exact c
  omit d

  include e
  definition cube_concat2 : cube (s₀₁₁ ⬝h s₀₃₁) (s₂₁₁ ⬝h s₂₃₁) s₁₀₁ s₁₄₁ (s₁₁₀ ⬝h s₁₃₀) (s₁₁₂ ⬝h s₁₃₂) :=
  by induction e; exact c
  omit e

  include f
  definition cube_concat3 : cube (s₀₁₁ ⬝v s₀₁₃) (s₂₁₁ ⬝v s₂₁₃) (s₁₀₁ ⬝v s₁₀₃) (s₁₂₁ ⬝v s₁₂₃) s₁₁₀ s₁₁₄ :=
  by induction f; exact c
  omit f c

  definition eq_of_cube (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    transpose s₁₀₁⁻¹ᵛ ⬝h s₁₁₀ ⬝h transpose s₁₂₁ =
      whisker_square (eq_bot_of_square s₀₁₁) (eq_bot_of_square s₂₁₁) idp idp s₁₁₂ :=
  by induction c; reflexivity

  definition eq_of_deg12_cube {s s' : square p₀₁₀ p₂₁₀ p₁₀₀ p₁₂₀}
    (c : cube vrfl vrfl vrfl vrfl s s') : s = s' :=
  by induction s; exact eq_of_cube c

  definition square_pathover [unfold 7]
    {f₁ : A → b₁ = b₂} {f₂ : A → b₃ = b₄} {f₃ : A → b₁ = b₃} {f₄ : A → b₂ = b₄}
    {p : a = a'}
    {q : square (f₁ a) (f₂ a) (f₃ a) (f₄ a)}
    {r : square (f₁ a') (f₂ a') (f₃ a') (f₄ a')}
    (s : cube (vdeg_square (ap f₁ p)) (vdeg_square (ap f₂ p))
              (vdeg_square (ap f₃ p)) (vdeg_square (ap f₄ p)) q r) : q =[p] r :=
  by induction p;apply pathover_idp_of_eq;exact eq_of_deg12_cube s

  /- Transporting along a square -/

  definition cube_transport110 {s₁₁₀' : square p₀₁₀ p₂₁₀ p₁₀₀ p₁₂₀}
    (p : s₁₁₀ = s₁₁₀') (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀' s₁₁₂ :=
  by induction p; exact c

  definition cube_transport112 {s₁₁₂' : square p₀₁₂ p₂₁₂ p₁₀₂ p₁₂₂}
    (p : s₁₁₂ = s₁₁₂') (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂':=
  by induction p; exact c

  definition cube_transport011 {s₀₁₁' : square p₀₁₀ p₀₁₂ p₀₀₁ p₀₂₁}
    (p : s₀₁₁ = s₀₁₁') (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    cube s₀₁₁' s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂ :=
  by induction p; exact c

  definition cube_transport211 {s₂₁₁' : square p₂₁₀ p₂₁₂ p₂₀₁ p₂₂₁}
    (p : s₂₁₁ = s₂₁₁') (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    cube s₀₁₁ s₂₁₁' s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂ :=
  by induction p; exact c

  definition cube_transport101 {s₁₀₁' : square p₁₀₀ p₁₀₂ p₀₀₁ p₂₀₁}
    (p : s₁₀₁ = s₁₀₁') (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    cube s₀₁₁ s₂₁₁ s₁₀₁' s₁₂₁ s₁₁₀ s₁₁₂ :=
  by induction p; exact c

  definition cube_transport121 {s₁₂₁' : square p₁₂₀ p₁₂₂ p₀₂₁ p₂₂₁}
    (p : s₁₂₁ = s₁₂₁') (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁' s₁₁₀ s₁₁₂ :=
  by induction p; exact c

  /- Each equality between squares leads to a cube which is degenerate in one
     dimension. -/

  definition deg1_cube {s₁₁₀' : square p₀₁₀ p₂₁₀ p₁₀₀ p₁₂₀} (p : s₁₁₀ = s₁₁₀') :
    cube s₁₁₀ s₁₁₀' hrfl hrfl vrfl vrfl :=
  by induction p; exact rfl1

  definition deg2_cube {s₁₁₀' : square p₀₁₀ p₂₁₀ p₁₀₀ p₁₂₀} (p : s₁₁₀ = s₁₁₀') :
    cube hrfl hrfl s₁₁₀ s₁₁₀' hrfl hrfl :=
  by induction p; exact rfl2

  definition deg3_cube {s₁₁₀' : square p₀₁₀ p₂₁₀ p₁₀₀ p₁₂₀} (p : s₁₁₀ = s₁₁₀') :
    cube vrfl vrfl vrfl vrfl s₁₁₀ s₁₁₀' :=
  by induction p; exact rfl3

  /- For each square of parralel equations, there are cubes where the square's
     sides appear in a degenerated way and two opposite sides are ids's -/

  section
  variables {a₀ a₁ : A} {p₀₀ p₀₂ p₂₀ p₂₂ : a₀ = a₁} {s₁₀ : p₀₀ = p₂₀}
    {s₁₂ : p₀₂ = p₂₂} {s₀₁ : p₀₀ = p₀₂} {s₂₁ : p₂₀ = p₂₂}
    (sq : square s₁₀ s₁₂ s₀₁ s₂₁)
  
  include sq
  
  definition ids3_cube_of_square : cube (hdeg_square s₀₁)
    (hdeg_square s₂₁) (hdeg_square s₁₀) (hdeg_square s₁₂) ids ids :=
  by induction p₀₀; induction sq; apply idc

  definition ids1_cube_of_square : cube ids ids
    (vdeg_square s₁₀) (vdeg_square s₁₂) (hdeg_square s₀₁) (hdeg_square s₂₁) :=
  by induction p₀₀; induction sq; apply idc

  definition ids2_cube_of_square : cube (vdeg_square s₁₀) (vdeg_square s₁₂)
    ids ids (vdeg_square s₀₁) (vdeg_square s₂₁) :=
  by induction p₀₀; induction sq; apply idc

  end

  /- Cube fillers -/

  section cube_fillers
  variables (s₁₁₀ s₁₁₂ s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁)

  definition cube_fill110 : Σ lid, cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ lid s₁₁₂ :=
  begin
    induction s₀₁₁, induction s₂₁₁,
    let fillsq := square_fill_l (eq_of_vdeg_square s₁₀₁) 
        (eq_of_hdeg_square s₁₁₂) (eq_of_vdeg_square s₁₂₁),
    apply sigma.mk,
    apply cube_transport101 (left_inv (vdeg_square_equiv _ _) s₁₀₁),
    apply cube_transport112 (left_inv (hdeg_square_equiv _ _) s₁₁₂),
    apply cube_transport121 (left_inv (vdeg_square_equiv _ _) s₁₂₁),
    apply ids1_cube_of_square, exact fillsq.2
  end

  definition cube_fill112 : Σ lid, cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ lid :=
  begin
    induction s₀₁₁, induction s₂₁₁,
    let fillsq := square_fill_r (eq_of_vdeg_square s₁₀₁) 
        (eq_of_hdeg_square s₁₁₀) (eq_of_vdeg_square s₁₂₁),
    apply sigma.mk,
    apply cube_transport101 (left_inv (vdeg_square_equiv _ _) s₁₀₁),
    apply cube_transport110 (left_inv (hdeg_square_equiv _ _) s₁₁₀),
    apply cube_transport121 (left_inv (vdeg_square_equiv _ _) s₁₂₁),
    apply ids1_cube_of_square, exact fillsq.2,
  end

  definition cube_fill011 : Σ lid, cube lid s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂ :=
  begin
    induction s₁₀₁, induction s₁₂₁,
    let fillsq := square_fill_t (eq_of_vdeg_square s₁₁₀) (eq_of_vdeg_square s₁₁₂)
      (eq_of_vdeg_square s₂₁₁),
    apply sigma.mk,
    apply cube_transport110 (left_inv (vdeg_square_equiv _ _) s₁₁₀),
    apply cube_transport211 (left_inv (vdeg_square_equiv _ _) s₂₁₁),
    apply cube_transport112 (left_inv (vdeg_square_equiv _ _) s₁₁₂),
    apply ids2_cube_of_square, exact fillsq.2,
  end

  definition cube_fill211 : Σ lid, cube s₀₁₁ lid s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂ :=
  begin
    induction s₁₀₁, induction s₁₂₁,
    let fillsq := square_fill_b (eq_of_vdeg_square s₀₁₁) (eq_of_vdeg_square s₁₁₀)
      (eq_of_vdeg_square s₁₁₂),
    apply sigma.mk,
    apply cube_transport011 (left_inv (vdeg_square_equiv _ _) s₀₁₁),
    apply cube_transport110 (left_inv (vdeg_square_equiv _ _) s₁₁₀),
    apply cube_transport112 (left_inv (vdeg_square_equiv _ _) s₁₁₂),
    apply ids2_cube_of_square, exact fillsq.2,
  end

  definition cube_fill101 : Σ lid, cube s₀₁₁ s₂₁₁ lid s₁₂₁ s₁₁₀ s₁₁₂ :=
  begin
    induction s₁₁₀, induction s₁₁₂,
    let fillsq := square_fill_t (eq_of_hdeg_square s₀₁₁) (eq_of_hdeg_square s₂₁₁)
      (eq_of_hdeg_square s₁₂₁),
    apply sigma.mk,
    apply cube_transport011 (left_inv (hdeg_square_equiv _ _) s₀₁₁),
    apply cube_transport211 (left_inv (hdeg_square_equiv _ _) s₂₁₁),
    apply cube_transport121 (left_inv (hdeg_square_equiv _ _) s₁₂₁),
    apply ids3_cube_of_square, exact fillsq.2,
  end

  definition cube_fill121 : Σ lid, cube s₀₁₁ s₂₁₁ s₁₀₁ lid s₁₁₀ s₁₁₂ :=
  begin
    induction s₁₁₀, induction s₁₁₂,
    let fillsq := square_fill_b (eq_of_hdeg_square s₁₀₁) (eq_of_hdeg_square s₀₁₁)
      (eq_of_hdeg_square s₂₁₁),
    apply sigma.mk,
    apply cube_transport101 (left_inv (hdeg_square_equiv _ _) s₁₀₁),
    apply cube_transport011 (left_inv (hdeg_square_equiv _ _) s₀₁₁),
    apply cube_transport211 (left_inv (hdeg_square_equiv _ _) s₂₁₁),
    apply ids3_cube_of_square, exact fillsq.2,
  end

  end cube_fillers

  /- Apply a non-dependent function to an entire cube -/

  include c
  definition apc (f : A → B) :
    cube (aps f s₀₁₁) (aps f s₂₁₁) (aps f s₁₀₁) (aps f s₁₂₁) (aps f s₁₁₀) (aps f s₁₁₂) :=
  by cases c; exact idc
  omit c

  /- Transpose a cube (swap dimensions) -/

  include c
  definition transpose12 : cube s₁₀₁ s₁₂₁ s₀₁₁ s₂₁₁ (transpose s₁₁₀) (transpose s₁₁₂) :=
  by cases c; exact idc

  definition transpose13 : cube s₁₁₀ s₁₁₂ (transpose s₁₀₁) (transpose s₁₂₁) s₀₁₁ s₂₁₁ :=
  by cases c; exact idc

  definition transpose23 : cube (transpose s₀₁₁) (transpose s₂₁₁) (transpose s₁₁₀)
    (transpose s₁₁₂) (transpose s₁₀₁) (transpose s₁₂₁) :=
  by cases c; exact idc
  omit c

  /- Inverting a cube along one dimension -/

  include c
  definition cube_inverse1 : cube s₂₁₁ s₀₁₁ s₁₀₁⁻¹ʰ s₁₂₁⁻¹ʰ s₁₁₀⁻¹ᵛ s₁₁₂⁻¹ᵛ :=
  by cases c; exact idc

  definition cube_inverse2 : cube s₀₁₁⁻¹ʰ s₂₁₁⁻¹ʰ s₁₂₁ s₁₀₁ s₁₁₀⁻¹ʰ s₁₁₂⁻¹ʰ :=
  by cases c; exact idc

  definition cube_inverse3 : cube s₀₁₁⁻¹ᵛ s₂₁₁⁻¹ᵛ s₁₀₁⁻¹ᵛ s₁₂₁⁻¹ᵛ s₁₁₂ s₁₁₀ :=
  by cases c; exact idc

  omit c

end eq