/-
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

open eq prod equiv pushout is_trunc bool

namespace join

  definition join (A B : Type) : 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 (A B : Type) [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 (A : Type) : 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 pushout.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 _) !pushout.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

end join