Spectral/homotopy/join_theorem.hlean

/-- Authors: Clive, Egbert --/

import homotopy.connectedness homotopy.join

open eq sigma pi function join is_conn is_trunc equiv is_equiv

namespace retraction
  variables {A B C : Type} (r2 : B → C) (r1 : A → B)


  definition is_retraction_compose
    [Hr2 : is_retraction r2] [Hr1 : is_retraction r1] :
    is_retraction (r2 ∘ r1) :=
  begin
    cases Hr2 with s2 s2_is_right_inverse,
    cases Hr1 with s1 s1_is_right_inverse,
    fapply is_retraction.mk,
    { exact s1 ∘ s2},
    { intro b, esimp,
      calc
      r2 (r1 (s1 (s2 (b)))) = r2 (s2 (b)) : ap r2 (s1_is_right_inverse (s2 b))
                        ... = b           : s2_is_right_inverse b

    }, /-- QED --/
  end

  definition is_retraction_compose_equiv_left [Hr2 : is_equiv r2] [Hr1 : is_retraction r1] : is_retraction (r2 ∘ r1) :=
  begin
    apply is_retraction_compose,
  end

  definition is_retraction_compose_equiv_right [Hr2 : is_retraction r2] [Hr1 :  is_equiv r1] : is_retraction (r2 ∘ r1) :=
  begin
    apply is_retraction_compose,
  end

end retraction

namespace is_conn
section

    open retraction

    universe variable u
    parameters (n : ℕ₋₂) {A : Type.{u}}
    parameter sec : ΠV : trunctype.{u} n,
                    is_retraction (const A : V → (A → V))

    include sec

    protected definition intro : is_conn n A :=
    begin
      apply is_conn_of_map_to_unit,
      apply is_conn_fun.intro,
      intro P,
      refine is_retraction_compose_equiv_right (const A) (pi_unit_left P),
    end
end
end is_conn

section Join_Theorem

variables (X Y : Type)
          (m n : ℕ₋₂)
          [HXm : is_conn m X]
          [HYn : is_conn n Y]

include HXm HYn

theorem is_conn_join : is_conn (m +2+ n) (join X Y) :=
begin
  apply is_conn.intro,
  intro V,
  apply is_retraction_of_is_equiv,
  apply is_equiv_of_is_contr_fun,
  intro f,
  refine is_contr_equiv_closed _ _,
  {exact unit},
  symmetry,
  exact sorry
end

end Join_Theorem