Spectral/homotopy/LES_applications.hlean

import .LES_of_homotopy_groups homotopy.connectedness homotopy.homotopy_group homotopy.join
open eq is_trunc pointed is_conn is_equiv fiber equiv trunc nat chain_complex prod fin algebra
     group trunc_index function join pushout

namespace nat
  open sigma sum
  definition eq_even_or_eq_odd (n : ℕ) : (Σk, 2 * k = n) ⊎ (Σk, 2 * k + 1 = n) :=
  begin
    induction n with n IH,
    { exact inl ⟨0, idp⟩},
    { induction IH with H H: induction H with k p: induction p,
      { exact inr ⟨k, idp⟩},
      { refine inl ⟨k+1, idp⟩}}
  end

  definition rec_on_even_odd {P : ℕ → Type} (n : ℕ) (H : Πk, P (2 * k)) (H2 : Πk, P (2 * k + 1))
    : P n :=
  begin
    cases eq_even_or_eq_odd n with v v: induction v with k p: induction p,
    { exact H k},
    { exact H2 k}
  end

end nat
open nat

namespace is_conn

  local attribute comm_group.to_group [coercion]
  local attribute is_equiv_tinverse [instance]

  theorem is_equiv_π_of_is_connected.{u} {A B : pType.{u}} (n k : ℕ) (f : A →* B)
    [H : is_conn_fun n f] (H2 : k ≤ n) : is_equiv (π→[k] f) :=
  begin
    induction k using rec_on_even_odd with k: cases k with k,
    { /- k = 0 -/
      change (is_equiv (trunc_functor 0 f)), apply is_equiv_trunc_functor_of_is_conn_fun,
      refine is_conn_fun_of_le f (zero_le_of_nat n)},
    { /- k > 0 even -/
      have H2' : 2 * k + 1 ≤ n, from le.trans !self_le_succ H2,
      exact
      @is_equiv_of_trivial _
        (LES_of_homotopy_groups3 f) _
        (is_exact_LES_of_homotopy_groups3 f (k, 5))
        (is_exact_LES_of_homotopy_groups3 f (succ k, 0))
        (@is_contr_HG_fiber_of_is_connected A B (2 * k + 1) n f H H2')
        (@is_contr_HG_fiber_of_is_connected A B (2 * succ k) n f H H2)
        (@pgroup_of_group _ (comm_group_LES_of_homotopy_groups3 f k 0) idp)
        (@pgroup_of_group _ (comm_group_LES_of_homotopy_groups3 f k 1) idp)
        (homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun3 f (k, 0)))},
    { /- k = 1 -/
      exact sorry},
    { /- k > 1 odd -/
      have H2' : 2 * succ k ≤ n, from le.trans !self_le_succ H2,
      have H3 : is_equiv (π→*[2*(succ k) + 1] f ∘* tinverse), from
      @is_equiv_of_trivial _
        (LES_of_homotopy_groups3 f) _
        (is_exact_LES_of_homotopy_groups3 f (succ k, 2))
        (is_exact_LES_of_homotopy_groups3 f (succ k, 3))
        (@is_contr_HG_fiber_of_is_connected A B (2 * succ k) n f H H2')
        (@is_contr_HG_fiber_of_is_connected A B (2 * succ k + 1) n f H H2)
        (@pgroup_of_group _ (comm_group_LES_of_homotopy_groups3 f k 3) idp)
        (@pgroup_of_group _ (comm_group_LES_of_homotopy_groups3 f k 4) idp)
        (homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun3 f (k, 3))),
      exact @(is_equiv.cancel_right tinverse) !is_equiv_tinverse
                  (pmap.to_fun (π→*[2*(succ k) + 1] f)) H3}
  end

  theorem is_surjective_π_of_is_connected.{u} {A B : pType.{u}} (n : ℕ) (f : A →* B)
    [H : is_conn_fun n f] : is_surjective (π→[n + 1] f) :=
  begin
    induction n using rec_on_even_odd with n,
    { have H3 : is_surjective (π→*[2*n + 1] f ∘* tinverse), from
      @is_surjective_of_trivial _
        (LES_of_homotopy_groups3 f) _
        (is_exact_LES_of_homotopy_groups3 f (n, 2))
        (@is_contr_HG_fiber_of_is_connected A B (2 * n) (2 * n) f H !le.refl),
      exact @(is_surjective_cancel_right (pmap.to_fun (π→*[2*n + 1] f)) tinverse) H3},
    { exact @is_surjective_of_trivial _
        (LES_of_homotopy_groups3 f) _
        (is_exact_LES_of_homotopy_groups3 f (k, 5))
        (@is_contr_HG_fiber_of_is_connected A B (2 * k + 1) (2 * k + 1) f H !le.refl)}
  end

  /- joins -/

  definition join_empty_right [constructor] (A : Type) : join A empty ≃ A :=
  begin
    fapply equiv.MK,
    { intro x, induction x with a o a o,
      { exact a },
      { exact empty.elim o },
      { exact empty.elim o } },
    { exact pushout.inl },
    { intro a, reflexivity},
    { intro x, induction x with a o a o,
      { reflexivity },
      { exact empty.elim o },
      { exact empty.elim o } }
  end

  definition natural_square2 {A B X : Type} {f : A → X} {g : B → X} (h : Πa b, f a = g b)
    {a a' : A} {b b' : B} (p : a = a') (q : b = b')
    : square (ap f p) (ap g q) (h a b) (h a' b') :=
  by induction p; induction q; exact hrfl

  section
    open sphere sphere_index

    definition add_plus_one_minus_one (n : ℕ₋₁) : n +1+ -1 = n := idp
    definition add_plus_one_succ (n m : ℕ₋₁) : n +1+ (m.+1) = (n +1+ m).+1 := idp
    definition minus_one_add_plus_one (n : ℕ₋₁) : -1 +1+ n = n :=
    begin induction n with n IH, reflexivity, exact ap succ IH end
    definition succ_add_plus_one (n m : ℕ₋₁) : (n.+1) +1+ m = (n +1+ m).+1 :=
    begin induction m with m IH, reflexivity, exact ap succ IH end

  end

end is_conn