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

Declaration of a join as a special case of a pushout
-/

import hit.pushout .susp cubical.cube cubical.squareover

open eq function prod equiv pushout is_trunc bool sigma.ops function

namespace join
  section
  variables (A B C : Type)

  definition join : Type := @pushout (A × B) A B pr1 pr2

  definition jglue {A B : Type} (a : A) (b : B) := @glue (A × B) A B pr1 pr2 (a, b)

  protected definition is_contr [HA : is_contr A] :
    is_contr (join A B) :=
  begin
    fapply is_contr.mk, exact inl (center A),
    intro x, induction x with a b, apply ap inl, apply center_eq,
    apply jglue, induction x with a b, apply pathover_of_tr_eq,
    apply concat, apply transport_eq_Fr, esimp, rewrite ap_id,
    generalize center_eq a, intro p, cases p, apply idp_con,
  end

  protected definition bool : join bool A ≃ susp A :=
  begin
    fapply equiv.MK, intro ba, induction ba with b a,
      induction b, exact susp.south, exact susp.north, exact susp.north,
      induction x with b a, esimp,
      induction b, apply inverse, apply susp.merid, exact a, reflexivity,
    intro s, induction s with m,
      exact inl tt, exact inl ff, exact (jglue tt m) ⬝ (jglue ff m)⁻¹,
    intros, induction b with m, do 2 reflexivity, esimp,
      apply eq_pathover, apply hconcat, apply hdeg_square, apply concat,
      apply ap_compose' (pushout.elim _ _ _), apply concat,
        apply ap (ap (pushout.elim _ _ _)), apply susp.elim_merid, apply ap_con,
      apply hconcat, apply vconcat, apply hdeg_square, apply elim_glue,
        apply hdeg_square, apply ap_inv, esimp,
      apply hconcat, apply hdeg_square, apply concat, apply idp_con,
        apply concat, apply ap inverse, apply elim_glue, apply inv_inv,
      apply hinverse, apply hdeg_square, apply ap_id,
    intro x, induction x with b a, induction b, do 2 reflexivity,
      esimp, apply jglue, induction x with b a, induction b, esimp,
      apply eq_pathover, rewrite ap_id,
      apply eq_hconcat, apply concat, apply ap_compose' (susp.elim _ _ _),
        apply concat, apply ap (ap _) !elim_glue,
        apply concat, apply ap_inv, 
        apply concat, apply ap inverse !susp.elim_merid,
        apply concat, apply con_inv, apply ap (λ x, x ⬝ _) !inv_inv,
      apply square_of_eq_top, apply inverse, 
        apply concat, apply ap (λ x, x ⬝ _) !con.assoc,
        rewrite [con.left_inv, con_idp], apply con.right_inv,
      esimp, apply eq_pathover, rewrite ap_id,
      apply eq_hconcat, apply concat, apply ap_compose' (susp.elim _ _ _),
        apply concat, apply ap (ap _) !elim_glue, esimp, reflexivity,
      apply square_of_eq_top, rewrite idp_con, apply !con.right_inv⁻¹,   
  end

  protected definition swap : join A B → join B A :=
  begin
    intro x, induction x with a b, exact inr a, exact inl b,
    apply !jglue⁻¹
  end

  protected definition swap_involutive (x : join A B) :
    join.swap B A (join.swap A B x) = x :=
  begin
    induction x with a b, do 2 reflexivity,
    induction x with a b, esimp,
    apply eq_pathover, rewrite ap_id,
    apply hdeg_square, esimp[join.swap],
    apply concat, apply ap_compose' (pushout.elim _ _ _),
    krewrite [elim_glue, ap_inv, elim_glue], apply inv_inv,
  end

  protected definition symm : join A B ≃ join B A :=
  begin
    fapply equiv.MK, do 2 apply join.swap,
    do 2 apply join.swap_involutive,
  end

  end

  --This proves that the join operator is associative
  --The proof is more or less ported from Evan Cavallo's agda version
  section join_switch
  private definition massage_sq {A : Type} {a₀₀ a₂₀ a₀₂ a₂₂ : A}
    {p₁₀ : a₀₀ = a₂₀} {p₁₂ :  a₀₂ = a₂₂} {p₀₁ : a₀₀ = a₀₂} {p₂₁ : a₂₀ = a₂₂}
    (sq : square p₁₀ p₁₂ p₀₁ p₂₁) : square p₁₀⁻¹ p₀₁⁻¹ (p₂₁ ⬝ p₁₂⁻¹) idp :=
  by induction sq; exact ids

  variables {A B C : Type}

  private definition switch_left [reducible] : join A B → join (join C B) A :=
  begin
    intro x, induction x with a b, exact inr a, exact inl (inr b), apply !jglue⁻¹,
  end

  private definition switch_coh_fill (a : A) (b : B) (c : C) :
    Σ sq : square (jglue (inl c) a)⁻¹ (ap inl (jglue c b)⁻¹) (ap switch_left (jglue a b)) idp,
      cube (hdeg_square !elim_glue) ids sq
        (!idp_con⁻¹ ⬝ph (massage_sq (square_Flr_ap_idp _ _)) ⬝vp !ap_inv⁻¹) hrfl hrfl :=
  by esimp; apply cube_fill101

  private definition switch_coh (ab : join A B) (c : C) : switch_left ab = inl (inl c) :=
  begin
    induction ab with a b, apply !jglue⁻¹, apply ap inl !jglue⁻¹, induction x with a b,
    apply eq_pathover, refine _ ⬝hp !ap_constant⁻¹,
    apply !switch_coh_fill.1,
  end

  protected definition switch [reducible] : join (join A B) C → join (join C B) A :=
  begin
    intro x, induction x with ab c, exact switch_left ab, exact inl (inl c),
    induction x with ab c, exact switch_coh ab c,
  end

  private definition switch_inv_left_square (a : A) (b : B) :
    square idp idp (ap (!(@join.switch C) ∘ switch_left) (jglue a b)) (ap inl (jglue a b)) :=
  begin
    refine hdeg_square !ap_compose ⬝h _,
    refine aps join.switch (hdeg_square !elim_glue) ⬝h _, esimp,
    refine hdeg_square !(ap_inv join.switch) ⬝h _,
    refine hrfl⁻¹ʰ⁻¹ᵛ ⬝h _, esimp[join.switch,switch_left,switch_coh],
    refine (hdeg_square !elim_glue)⁻¹ᵛ ⬝h _, esimp,
    refine (hdeg_square !ap_inv)⁻¹ᵛ ⬝h _, apply hdeg_square !inv_inv,
  end

  private definition switch_inv_coh_left (c : C) (a : A) :
    square idp idp (ap !(@join.switch C B) (switch_coh (inl a) c)) (jglue (inl a) c) :=
  begin
    refine hrfl ⬝h _,
    refine aps join.switch hrfl ⬝h _, esimp[switch_coh],
    refine hdeg_square !ap_inv ⬝h _,
    refine hrfl⁻¹ʰ⁻¹ᵛ ⬝h _, esimp[join.switch,switch_left],
    refine (hdeg_square !elim_glue)⁻¹ᵛ ⬝h _,
    refine hrfl⁻¹ᵛ ⬝h _, apply hdeg_square !inv_inv,
  end

  private definition switch_inv_coh_right (c : C) (b : B) :
    square idp idp (ap !(@join.switch _ _ A) (switch_coh (inr b) c)) (jglue (inr b) c) :=
  begin
    refine hrfl ⬝h _,
    refine aps join.switch hrfl ⬝h _, esimp[switch_coh],
    refine aps join.switch (hdeg_square !ap_inv) ⬝h _,
    refine hdeg_square !ap_inv ⬝h _,
    refine (hdeg_square !ap_compose)⁻¹ᵛ⁻¹ʰ ⬝h _,
    refine hrfl⁻¹ᵛ ⬝h _, esimp[join.switch,switch_left],
    refine (hdeg_square !elim_glue)⁻¹ᵛ ⬝h _, apply hdeg_square !inv_inv,
  end

  private definition switch_inv_left (ab : join A B) :
    !(@join.switch C) (join.switch (inl ab)) = inl ab :=
  begin
    induction ab with a b, do 2 reflexivity,
    induction x with a b, apply eq_pathover, exact !switch_inv_left_square,
  end

  section
  variables (a : A) (b : B) (c : C)

  private definition switch_inv_cube_aux1 {A B C : Type} {b : B} {f : A → B} (h : B → C)
    (g : Π a, f a = b) {x y : A} (p : x = y) :
    cube (hdeg_square (ap_compose h f p)) ids (square_Flr_ap_idp (λ a, ap h (g a)) p)
    (aps h (square_Flr_ap_idp _ _)) hrfl hrfl :=
  begin
    cases p, esimp[square_Flr_ap_idp], apply deg2_cube,
    cases (g x), reflexivity,
  end

  private definition switch_inv_cube_aux2 {A B C : Type} {b : B} {f : A → B}
    (g : Π a, f a = b) {x y : A} (p : x = y) {sq : square (g x) (g y) (ap f p) idp}
    (q : apdo g p = eq_pathover (sq ⬝hp !ap_constant⁻¹)) : square_Flr_ap_idp _ _ = sq :=
  begin
    cases p, esimp at *, exact sorry
  end

  private definition switch_inv_cube (a : A) (b : B) (c : C) :
    cube (switch_inv_left_square a b) ids (square_Flr_ap_idp _ _)
    (square_Flr_ap_idp _ _) (switch_inv_coh_left c a) (switch_inv_coh_right c b) :=
  begin
    esimp [switch_inv_coh_left, switch_inv_coh_right, switch_inv_left_square],
    apply cube_concat2, apply switch_inv_cube_aux1,
    apply cube_concat2, apply cube_transport101,
      apply inverse, apply ap (λ x, aps join.switch x),
      apply switch_inv_cube_aux2, apply rec_glue,
      apply apc, apply (switch_coh_fill a b c).2,
    apply cube_concat2, esimp, 
  end

  end

  private definition pathover_of_triangle_cube {A B : Type} {b₀ b₁ : A → B}
    {b : B} {p₀₁ : Π a, b₀ a = b₁ a} {p₀ : Π a, b₀ a = b} {p₁ : Π a, b₁ a = b}
    {x y : A} {q : x = y} {sqx : square (p₀₁ x) idp (p₀ x) (p₁ x)}
    {sqy : square (p₀₁ y) idp (p₀ y) (p₁ y)}
    (c : cube (natural_square_tr _ _) ids (square_Flr_ap_idp p₀ q) (square_Flr_ap_idp p₁ q)
       sqx sqy) :
    sqx =[q] sqy :=
  begin
    cases q, esimp [square_Flr_ap_idp] at *,
    apply pathover_of_eq_tr, esimp, apply eq_of_deg12_cube, exact c,
  end

  private definition pathover_of_ap_ap_square {A : Type} {x y : A} {p : x = y}
    (g : B → A) (f : A → B) {u : g (f x) = x} {v : g (f y) = y}
    (sq : square (ap g (ap f p)) p u v) : u =[p] v :=
  by cases p; apply eq_pathover; apply transpose; exact sq


  private definition hdeg_square_idp {A : Type} {a a' : A} {p : a = a'} :
    hdeg_square (refl p) = hrfl :=
  by cases p; reflexivity

  private definition vdeg_square_idp {A : Type} {a a' : A} {p : a = a'} :
    vdeg_square (refl p) = vrfl :=
  by cases p; reflexivity

  private definition natural_square_tr_beta {A B : Type} {f₁ f₂ : A → B}
    (p : Π a, f₁ a = f₂ a) {x y : A} (q : x = y) {sq : square (p x) (p y) (ap f₁ q) (ap f₂ q)}
    (e : apdo p q = eq_pathover sq) :
    natural_square_tr p q = sq :=
  begin
    cases q, esimp at *,
    apply concat, apply inverse, apply vdeg_square_idp,
    assert H : refl (p y) = eq_of_vdeg_square sq,
    { exact sorry },
    apply concat, apply ap vdeg_square, exact H,
    apply is_equiv.left_inv (equiv.to_fun !vdeg_square_equiv),
  end

  private definition switch_inv_coh (c : C) (k : join A B) :
    square (switch_inv_left k) idp (ap join.switch (switch_coh k c)) (jglue k c) :=
  begin
    induction k, apply switch_inv_coh_left, apply switch_inv_coh_right,
    refine pathover_of_triangle_cube _,
    induction x with [a, b], esimp, apply cube_transport011,
    apply inverse, rotate 1, apply switch_inv_cube,
    apply natural_square_tr_beta, apply rec_glue,
  end

  protected definition switch_involutive (x : join (join A B) C) :
    join.switch (join.switch x) = x :=
  begin
    induction x, apply switch_inv_left, reflexivity,
    apply pathover_of_ap_ap_square join.switch join.switch,
    induction x with [k, c], krewrite elim_glue, esimp,
    apply transpose, exact !switch_inv_coh,
  end

  end join_switch 

  protected definition switch_equiv (A B C : Type) :
    join (join A B) C ≃ join (join C B) A :=
  by apply equiv.MK; do 2 apply join.switch_involutive

  protected definition assoc (A B C : Type) :
    join (join A B) C ≃ join A (join B C) :=
  calc join (join A B) C ≃ join (join C B) A : join.switch_equiv
                     ... ≃ join A (join C B) : join.symm
                     ... ≃ join A (join B C) : join.symm

end join