Spectral/move_to_lib.hlean

-- definitions, theorems and attributes which should be moved to files in the HoTT library

import homotopy.sphere2 homotopy.cofiber homotopy.wedge

open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc_index pi group
     is_trunc function sphere unit sum prod bool

definition add_comm_right {A : Type} [add_comm_semigroup A] (n m k : A) : n + m + k = n + k + m :=
!add.assoc ⬝ ap (add n) !add.comm ⬝ !add.assoc⁻¹

structure is_exact_t {A B : Type} {C : Type*} (f : A → B) (g : B → C) :=
  ( im_in_ker : Π(a:A), g (f a) = pt)
  ( ker_in_im : Π(b:B), (g b = pt) → fiber f b)

structure is_exact {A B : Type} {C : Type*} (f : A → B) (g : B → C) :=
  ( im_in_ker : Π(a:A), g (f a) = pt)
  ( ker_in_im : Π(b:B), (g b = pt) → image f b)

definition is_exact_g {A B C : Group} (f : A →g B) (g : B →g C) :=
is_exact f g

definition is_exact_ag {A B C : AbGroup} (f : A →g B) (g : B →g C) :=
is_exact f g

definition is_exact_g.mk {A B C : Group} {f : A →g B} {g : B →g C}
  (H₁ : Πa, g (f a) = 1) (H₂ : Πb, g b = 1 → image f b) : is_exact_g f g :=
is_exact.mk H₁ H₂

namespace algebra
  definition inf_group_loopn (n : ℕ) (A : Type*) [H : is_succ n] : inf_group (Ω[n] A) :=
  by induction H; exact _

  definition one_unique {A : Type} [group A] {a : A} (H : Πb, a * b = b) : a = 1 :=
  !mul_one⁻¹ ⬝ H 1
end algebra

namespace eq

section -- squares
  variables {A B : Type} {a a' a'' a₀₀ a₂₀ a₄₀ a₀₂ a₂₂ a₂₄ a₀₄ a₄₂ a₄₄ a₁ a₂ a₃ a₄ : A}
            /-a₀₀-/ {p₁₀ p₁₀' : a₀₀ = a₂₀} /-a₂₀-/ {p₃₀ : a₂₀ = a₄₀} /-a₄₀-/
       {p₀₁ p₀₁' : a₀₀ = a₀₂} /-s₁₁-/ {p₂₁ p₂₁' : a₂₀ = a₂₂} /-s₃₁-/ {p₄₁ : a₄₀ = a₄₂}
            /-a₀₂-/ {p₁₂ p₁₂' : a₀₂ = a₂₂} /-a₂₂-/ {p₃₂ : a₂₂ = a₄₂} /-a₄₂-/
       {p₀₃ : a₀₂ = a₀₄} /-s₁₃-/ {p₂₃ : a₂₂ = a₂₄} /-s₃₃-/ {p₄₃ : a₄₂ = a₄₄}
            /-a₀₄-/ {p₁₄ : a₀₄ = a₂₄} /-a₂₄-/ {p₃₄ : a₂₄ = a₄₄} /-a₄₄-/

  variables {s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁} {s₃₁ : square p₃₀ p₃₂ p₂₁ p₄₁}
            {s₁₃ : square p₁₂ p₁₄ p₀₃ p₂₃} {s₃₃ : square p₃₂ p₃₄ p₂₃ p₄₃}

  definition natural_square_eq {A B : Type} {a a' : A} {f g : A → B} (p : f ~ g) (q : a = a')
    : natural_square p q = square_of_pathover (apd p q) :=
  idp

  definition eq_of_square_hrfl_hconcat_eq {A : Type} {a a' : A} {p p' : a = a'} (q : p = p')
    : eq_of_square (hrfl ⬝hp q⁻¹) = !idp_con ⬝ q :=
  by induction q; induction p; reflexivity

  definition aps_vrfl {A B : Type} {a a' : A} (f : A → B) (p : a = a') :
    aps f (vrefl p) = vrefl (ap f p) :=
  by induction p; reflexivity

  definition aps_hrfl {A B : Type} {a a' : A} (f : A → B) (p : a = a') :
    aps f (hrefl p) = hrefl (ap f p) :=
  by induction p; reflexivity

  -- should the following two equalities be cubes?
  definition natural_square_ap_fn {A B C : Type} {a a' : A} {g h : A → B} (f : B → C) (p : g ~ h)
    (q : a = a') : natural_square (λa, ap f (p a)) q =
      ap_compose f g q ⬝ph (aps f (natural_square p q) ⬝hp (ap_compose f h q)⁻¹) :=
  begin
    induction q, exact !aps_vrfl⁻¹
  end

  definition natural_square_compose {A B C : Type} {a a' : A} {g g' : B → C}
    (p : g ~ g') (f : A → B) (q : a = a') : natural_square (λa, p (f a)) q =
    ap_compose g f q ⬝ph (natural_square p (ap f q) ⬝hp (ap_compose g' f q)⁻¹) :=
  by induction q; reflexivity

  definition natural_square_eq2 {A B : Type} {a a' : A} {f f' : A → B} (p : f ~ f') {q q' : a = a'}
    (r : q = q') : natural_square p q = ap02 f r ⬝ph (natural_square p q' ⬝hp (ap02 f' r)⁻¹) :=
  by induction r; reflexivity

  definition natural_square_refl {A B : Type} {a a' : A} (f : A → B) (q : a = a')
    : natural_square (homotopy.refl f) q = hrfl :=
  by induction q; reflexivity

  definition aps_eq_hconcat {p₀₁'} (f : A → B) (q : p₀₁' = p₀₁) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) :
    aps f (q ⬝ph s₁₁) = ap02 f q ⬝ph aps f s₁₁ :=
  by induction q; reflexivity

  definition aps_hconcat_eq {p₂₁'} (f : A → B) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₂₁' = p₂₁) :
    aps f (s₁₁ ⬝hp r⁻¹) = aps f s₁₁ ⬝hp (ap02 f r)⁻¹ :=
  by induction r; reflexivity

  definition aps_hconcat_eq' {p₂₁'} (f : A → B) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₂₁ = p₂₁') :
    aps f (s₁₁ ⬝hp r) = aps f s₁₁ ⬝hp ap02 f r :=
  by induction r; reflexivity

  definition aps_square_of_eq (f : A → B) (s : p₁₀ ⬝ p₂₁ = p₀₁ ⬝ p₁₂) :
    aps f (square_of_eq s) = square_of_eq ((ap_con f p₁₀ p₂₁)⁻¹ ⬝ ap02 f s ⬝ ap_con f p₀₁ p₁₂) :=
  by induction p₁₂; esimp at *; induction s; induction p₂₁; induction p₁₀; reflexivity

  definition aps_eq_hconcat_eq {p₀₁' p₂₁'} (f : A → B) (q : p₀₁' = p₀₁) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
    (r : p₂₁' = p₂₁) : aps f (q ⬝ph s₁₁ ⬝hp r⁻¹) = ap02 f q ⬝ph aps f s₁₁ ⬝hp (ap02 f r)⁻¹ :=
  by induction q; induction r; reflexivity

end

section -- cubes

  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}
            {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}
            {b₁ b₂ b₃ b₄ : B}
            (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂)

  definition whisker001 {p₀₀₁' : a₀₀₀ = a₀₀₂} (q : p₀₀₁' = p₀₀₁)
    (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) : cube (q ⬝ph s₀₁₁) s₂₁₁ (q ⬝ph s₁₀₁) s₁₂₁ s₁₁₀ s₁₁₂ :=
  by induction q; exact c

  definition whisker021 {p₀₂₁' : a₀₂₀ = a₀₂₂} (q : p₀₂₁' = p₀₂₁)
    (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    cube (s₀₁₁ ⬝hp q⁻¹) s₂₁₁ s₁₀₁ (q ⬝ph s₁₂₁) s₁₁₀ s₁₁₂ :=
  by induction q; exact c

  definition whisker021' {p₀₂₁' : a₀₂₀ = a₀₂₂} (q : p₀₂₁ = p₀₂₁')
    (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    cube (s₀₁₁ ⬝hp q) s₂₁₁ s₁₀₁ (q⁻¹ ⬝ph s₁₂₁) s₁₁₀ s₁₁₂ :=
  by induction q; exact c

  definition whisker201 {p₂₀₁' : a₂₀₀ = a₂₀₂} (q : p₂₀₁' = p₂₀₁)
    (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    cube s₀₁₁ (q ⬝ph s₂₁₁) (s₁₀₁ ⬝hp q⁻¹) s₁₂₁ s₁₁₀ s₁₁₂ :=
  by induction q; exact c

  definition whisker201' {p₂₀₁' : a₂₀₀ = a₂₀₂} (q : p₂₀₁ = p₂₀₁')
    (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    cube s₀₁₁ (q⁻¹ ⬝ph s₂₁₁) (s₁₀₁ ⬝hp q) s₁₂₁ s₁₁₀ s₁₁₂ :=
  by induction q; exact c

  definition whisker221 {p₂₂₁' : a₂₂₀ = a₂₂₂} (q : p₂₂₁ = p₂₂₁')
    (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) : cube s₀₁₁ (s₂₁₁ ⬝hp q) s₁₀₁ (s₁₂₁ ⬝hp q) s₁₁₀ s₁₁₂ :=
  by induction q; exact c

  definition move221 {p₂₂₁' : a₂₂₀ = a₂₂₂} {s₁₂₁ : square p₁₂₀ p₁₂₂ p₀₂₁ p₂₂₁'} (q : p₂₂₁ = p₂₂₁')
    (c : cube s₀₁₁ (s₂₁₁ ⬝hp q) s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    cube s₀₁₁ s₂₁₁ s₁₀₁ (s₁₂₁ ⬝hp q⁻¹) s₁₁₀ s₁₁₂ :=
  by induction q; exact c

  definition move201 {p₂₀₁' : a₂₀₀ = a₂₀₂} {s₁₀₁ : square p₁₀₀ p₁₀₂ p₀₀₁ p₂₀₁'}  (q : p₂₀₁' = p₂₀₁)
    (c : cube s₀₁₁ (q ⬝ph s₂₁₁) s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
    cube s₀₁₁ s₂₁₁ (s₁₀₁ ⬝hp q) s₁₂₁ s₁₁₀ s₁₁₂ :=
  by induction q; exact c

end

  definition ap_eq_ap010 {A B C : Type} (f : A → B → C) {a a' : A} (p : a = a') (b : B) :
    ap (λa, f a b) p = ap010 f p b :=
  by reflexivity

  definition ap011_idp {A B C : Type} (f : A → B → C) {a a' : A} (p : a = a') (b : B) :
    ap011 f p idp = ap010 f p b :=
  by reflexivity

  definition ap011_flip {A B C : Type} (f : A → B → C) {a a' : A} {b b' : B} (p : a = a') (q : b = b') :
    ap011 f p q = ap011 (λb a, f a b) q p :=
  by induction q; induction p; reflexivity

  theorem apd_constant' {A A' : Type} {B : A' → Type} {a₁ a₂ : A} {a' : A'} (b : B a')
    (p : a₁ = a₂) : apd (λx, b) p = pathover_of_eq p idp :=
  by induction p; reflexivity

  definition apo011 {A : Type} {B C D : A → Type} {a a' : A} {p : a = a'} {b : B a} {b' : B a'}
    {c : C a} {c' : C a'} (f : Π⦃a⦄, B a → C a → D a) (q : b =[p] b') (r : c =[p] c') :
    f b c =[p] f b' c' :=
  begin induction q, induction r using idp_rec_on, exact idpo end

  definition ap011_ap_square_right {A B C : Type} (f : A → B → C) {a a' : A} (p : a = a')
    {b₁ b₂ b₃ : B} {q₁₂ : b₁ = b₂} {q₂₃ : b₂ = b₃} {q₁₃ : b₁ = b₃} (r : q₁₂ ⬝ q₂₃ = q₁₃) :
    square (ap011 f p q₁₂) (ap (λx, f x b₃) p) (ap (f a) q₁₃) (ap (f a') q₂₃) :=
  by induction r; induction q₂₃; induction q₁₂; induction p; exact ids

  definition ap011_ap_square_left {A B C : Type} (f : B → A → C) {a a' : A} (p : a = a')
    {b₁ b₂ b₃ : B} {q₁₂ : b₁ = b₂} {q₂₃ : b₂ = b₃} {q₁₃ : b₁ = b₃} (r : q₁₂ ⬝ q₂₃ = q₁₃) :
    square (ap011 f q₁₂ p) (ap (f b₃) p) (ap (λx, f x a) q₁₃) (ap (λx, f x a') q₂₃) :=
  by induction r; induction q₂₃; induction q₁₂; induction p; exact ids

  definition ap_ap011 {A B C D : Type} (g : C → D) (f : A → B → C) {a a' : A} {b b' : B}
    (p : a = a') (q : b = b') : ap g (ap011 f p q) = ap011 (λa b, g (f a b)) p q :=
  begin
    induction p, exact (ap_compose g (f a) q)⁻¹
  end

  definition con2_assoc {A : Type} {x y z t : A} {p p' : x = y} {q q' : y = z} {r r' : z = t}
    (h : p = p') (h' : q = q') (h'' : r = r') :
    square ((h ◾ h') ◾ h'') (h ◾ (h' ◾ h'')) (con.assoc p q r) (con.assoc p' q' r') :=
  by induction h; induction h'; induction h''; exact hrfl

  definition con_left_inv_idp {A : Type} {x : A} {p : x = x} (q : p = idp)
    : con.left_inv p = q⁻² ◾ q :=
  by cases q; reflexivity

  definition eckmann_hilton_con2 {A : Type} {x : A} {p p' q q': idp = idp :> x = x}
    (h : p = p') (h' : q = q') : square (h ◾ h') (h' ◾ h) (eckmann_hilton p q) (eckmann_hilton p' q') :=
  by induction h; induction h'; exact hrfl

  definition ap_con_fn {A B : Type} {a a' : A} {b : B} (g h : A → b = b) (p : a = a') :
    ap (λa, g a ⬝ h a) p = ap g p ◾ ap h p :=
  by induction p; reflexivity

  protected definition homotopy.rfl [reducible] [unfold_full] {A B : Type} {f : A → B} : f ~ f :=
  homotopy.refl f

  definition compose_id {A B : Type} (f : A → B) : f ∘ id ~ f :=
  by reflexivity

  definition id_compose {A B : Type} (f : A → B) : id ∘ f ~ f :=
  by reflexivity

  -- move to eq2
  definition ap_eq_ap011 {A B C X : Type} (f : A → B → C) (g : X → A) (h : X → B) {x x' : X}
    (p : x = x') : ap (λx, f (g x) (h x)) p = ap011 f (ap g p) (ap h p) :=
  by induction p; reflexivity

  definition ap_is_weakly_constant {A B : Type} {f : A → B}
    (h : is_weakly_constant f) {a a' : A} (p : a = a') : ap f p = (h a a)⁻¹ ⬝ h a a' :=
  by induction p; exact !con.left_inv⁻¹

  definition ap_is_constant_idp {A B : Type} {f : A → B} {b : B} (p : Πa, f a = b) {a : A} (q : a = a)
    (r : q = idp) : ap_is_constant p q = ap02 f r ⬝ (con.right_inv (p a))⁻¹ :=
  by cases r; exact !idp_con⁻¹

  definition con_right_inv_natural {A : Type} {a a' : A} {p p' : a = a'} (q : p = p') :
    con.right_inv p = q ◾ q⁻² ⬝ con.right_inv p' :=
  by induction q; induction p; reflexivity

  definition whisker_right_ap {A B : Type} {a a' : A}{b₁ b₂ b₃ : B} (q : b₂ = b₃) (f : A → b₁ = b₂)
    (p : a = a') : whisker_right q (ap f p) = ap (λa, f a ⬝ q) p :=
  by induction p; reflexivity

  infix ` ⬝hty `:75 := homotopy.trans
  postfix `⁻¹ʰᵗʸ`:(max+1) := homotopy.symm

  definition hassoc {A B C D : Type} (h : C → D) (g : B → C) (f : A → B) : (h ∘ g) ∘ f ~ h ∘ (g ∘ f) :=
  λa, idp

  -- to algebra.homotopy_group
  definition homotopy_group_homomorphism_pcompose (n : ℕ) [H : is_succ n] {A B C : Type*} (g : B →* C)
    (f : A →* B) : π→g[n] (g ∘* f) ~ π→g[n] g ∘ π→g[n] f :=
  begin
    induction H with n, exact to_homotopy (homotopy_group_functor_compose (succ n) g f)
  end

  definition apn_pinv (n : ℕ) {A B : Type*} (f : A ≃* B) :
    Ω→[n] f⁻¹ᵉ* ~* (loopn_pequiv_loopn n f)⁻¹ᵉ* :=
  begin
    refine !to_pinv_pequiv_MK2⁻¹*
  end

  -- definition homotopy_group_homomorphism_pinv (n : ℕ) {A B : Type*} (f : A ≃* B) :
  --   π→g[n+1] f⁻¹ᵉ* ~ (homotopy_group_isomorphism_of_pequiv n f)⁻¹ᵍ :=
  -- begin
  --   -- refine ptrunc_functor_phomotopy 0 !apn_pinv ⬝hty _,
  --   -- intro x, esimp,
  -- end

  -- definition natural_square_tr_eq {A B : Type} {a a' : A} {f g : A → B}
  --   (p : f ~ g) (q : a = a') : natural_square p q = square_of_pathover (apd p q) :=
  -- idp


  section hsquare
  variables {A₀₀ A₂₀ A₄₀ A₀₂ A₂₂ A₄₂ A₀₄ A₂₄ A₄₄ : Type}
            {f₁₀ : A₀₀ → A₂₀} {f₃₀ : A₂₀ → A₄₀}
            {f₀₁ : A₀₀ → A₀₂} {f₂₁ : A₂₀ → A₂₂} {f₄₁ : A₄₀ → A₄₂}
            {f₁₂ : A₀₂ → A₂₂} {f₃₂ : A₂₂ → A₄₂}
            {f₀₃ : A₀₂ → A₀₄} {f₂₃ : A₂₂ → A₂₄} {f₄₃ : A₄₂ → A₄₄}
            {f₁₄ : A₀₄ → A₂₄} {f₃₄ : A₂₄ → A₄₄}

  definition hsquare [reducible] (f₁₀ : A₀₀ → A₂₀) (f₁₂ : A₀₂ → A₂₂)
                                 (f₀₁ : A₀₀ → A₀₂) (f₂₁ : A₂₀ → A₂₂) : Type :=
  f₂₁ ∘ f₁₀ ~ f₁₂ ∘ f₀₁

  definition hsquare_of_homotopy (p : f₂₁ ∘ f₁₀ ~ f₁₂ ∘ f₀₁) : hsquare f₁₀ f₁₂ f₀₁ f₂₁ :=
  p

  definition homotopy_of_hsquare (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) : f₂₁ ∘ f₁₀ ~ f₁₂ ∘ f₀₁ :=
  p

  definition hhconcat (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) (q : hsquare f₃₀ f₃₂ f₂₁ f₄₁) :
    hsquare (f₃₀ ∘ f₁₀) (f₃₂ ∘ f₁₂) f₀₁ f₄₁ :=
  hwhisker_right f₁₀ q ⬝hty hwhisker_left f₃₂ p

  definition hvconcat (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) (q : hsquare f₁₂ f₁₄ f₀₃ f₂₃) :
    hsquare f₁₀ f₁₄ (f₀₃ ∘ f₀₁) (f₂₃ ∘ f₂₁) :=
  (hhconcat p⁻¹ʰᵗʸ q⁻¹ʰᵗʸ)⁻¹ʰᵗʸ

  definition hhinverse {f₁₀ : A₀₀ ≃ A₂₀} {f₁₂ : A₀₂ ≃ A₂₂} (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) :
    hsquare f₁₀⁻¹ᵉ f₁₂⁻¹ᵉ f₂₁ f₀₁ :=
  λb, eq_inv_of_eq ((p (f₁₀⁻¹ᵉ b))⁻¹ ⬝ ap f₂₁ (to_right_inv f₁₀ b))

  definition hvinverse {f₀₁ : A₀₀ ≃ A₀₂} {f₂₁ : A₂₀ ≃ A₂₂} (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) :
    hsquare f₁₂ f₁₀ f₀₁⁻¹ᵉ f₂₁⁻¹ᵉ :=
  (hhinverse p⁻¹ʰᵗʸ)⁻¹ʰᵗʸ

  infix ` ⬝htyh `:73 := hhconcat
  infix ` ⬝htyv `:73 := hvconcat
  postfix `⁻¹ʰᵗʸʰ`:(max+1) := hhinverse
  postfix `⁻¹ʰᵗʸᵛ`:(max+1) := hvinverse

  end hsquare
  -- move to init.funext
  protected definition homotopy.rec_on_idp_left [recursor] {A : Type} {P : A → Type} {g : Πa, P a}
    {Q : Πf, (f ~ g) → Type} {f : Π x, P x}
    (p : f ~ g) (H : Q g (homotopy.refl g)) : Q f p :=
  begin
    induction p using homotopy.rec_on, induction q, exact H
  end

  --eq2 (duplicate of ap_compose_ap02_constant)
  definition ap02_ap_constant {A B C : Type} {a a' : A} (f : B → C) (b : B) (p : a = a') :
    square (ap_constant p (f b)) (ap02 f (ap_constant p b)) (ap_compose f (λx, b) p) idp :=
  by induction p; exact ids

  definition ap_constant_compose {A B C : Type} {a a' : A} (c : C) (f : A → B) (p : a = a') :
    square (ap_constant p c) (ap_constant (ap f p) c) (ap_compose (λx, c) f p) idp :=
  by induction p; exact ids

  definition ap02_constant {A B : Type} {a a' : A} (b : B) {p p' : a = a'}
    (q : p = p') : square (ap_constant p b) (ap_constant p' b) (ap02 (λx, b) q) idp :=
  by induction q; exact vrfl

end eq open eq

namespace wedge
  open pushout unit
  protected definition glue (A B : Type*) : inl pt = inr pt :> wedge A B :=
  pushout.glue ⋆

end wedge

namespace pi

  definition is_contr_pi_of_neg {A : Type} (B : A → Type) (H : ¬ A) : is_contr (Πa, B a) :=
  begin
    apply is_contr.mk (λa, empty.elim (H a)), intro f, apply eq_of_homotopy, intro x, contradiction
  end
end pi

namespace trunc

  -- TODO: redefine loopn_ptrunc_pequiv
  definition apn_ptrunc_functor (n : ℕ₋₂) (k : ℕ) {A B : Type*} (f : A →* B) :
    Ω→[k] (ptrunc_functor (n+k) f) ∘* (loopn_ptrunc_pequiv n k A)⁻¹ᵉ* ~*
    (loopn_ptrunc_pequiv n k B)⁻¹ᵉ* ∘* ptrunc_functor n (Ω→[k] f) :=
  begin
    revert n, induction k with k IH: intro n,
    { reflexivity },
    { exact sorry }
  end

  definition ptrunc_pequiv_natural [constructor] (n : ℕ₋₂) {A B : Type*} (f : A →* B) [is_trunc n A]
    [is_trunc n B] : f ∘* ptrunc_pequiv n A ~* ptrunc_pequiv n B ∘* ptrunc_functor n f :=
  begin
    fapply phomotopy.mk,
    { intro a, induction a with a, reflexivity },
    { refine !idp_con ⬝ _ ⬝ !idp_con⁻¹, refine !ap_compose'⁻¹ ⬝ _, apply ap_id }
  end

  definition ptr_natural [constructor] (n : ℕ₋₂) {A B : Type*} (f : A →* B) :
    ptrunc_functor n f ∘* ptr n A ~* ptr n B ∘* f :=
  begin
    fapply phomotopy.mk,
    { intro a, reflexivity },
    { reflexivity }
  end

  definition ptrunc_elim_pcompose (n : ℕ₋₂) {A B C : Type*} (g : B →* C) (f : A →* B) [is_trunc n B]
    [is_trunc n C] : ptrunc.elim n (g ∘* f) ~* g ∘* ptrunc.elim n f :=
  begin
    fapply phomotopy.mk,
    { intro a, induction a with a, reflexivity },
    { apply idp_con }
  end

end trunc

namespace is_equiv

  definition inv_homotopy_inv {A B : Type} {f g : A → B} [is_equiv f] [is_equiv g] (p : f ~ g)
    : f⁻¹ ~ g⁻¹ :=
  λb, (left_inv g (f⁻¹ b))⁻¹ ⬝ ap g⁻¹ ((p (f⁻¹ b))⁻¹ ⬝ right_inv f b)

  definition to_inv_homotopy_to_inv {A B : Type} {f g : A ≃ B} (p : f ~ g) : f⁻¹ᵉ ~ g⁻¹ᵉ :=
  inv_homotopy_inv p

end is_equiv

namespace prod

  definition pprod_functor [constructor] {A B C D : Type*} (f : A →* C) (g : B →* D) : A ×* B →* C ×* D :=
  pmap.mk (prod_functor f g) (prod_eq (respect_pt f) (respect_pt g))

  open prod.ops
  definition prod_pathover_equiv {A : Type} {B C : A → Type} {a a' : A} (p : a = a')
    (x : B a × C a) (x' : B a' × C a') : x =[p] x' ≃ x.1 =[p] x'.1 × x.2 =[p] x'.2 :=
  begin
    fapply equiv.MK,
    { intro q, induction q, constructor: constructor },
    { intro v, induction v with q r, exact prod_pathover _ _ _ q r },
    { intro v, induction v with q r, induction x with b c, induction x' with b' c',
      esimp at *, induction q, refine idp_rec_on r _, reflexivity },
    { intro q, induction q, induction x with b c, reflexivity }
  end

end prod open prod

namespace sigma

  -- set_option pp.notation false
  -- set_option pp.binder_types true

  open sigma.ops
  definition pathover_pr1 [unfold 9] {A : Type} {B : A → Type} {C : Πa, B a → Type}
    {a a' : A} {p : a = a'} {x : Σb, C a b} {x' : Σb', C a' b'}
    (q : x =[p] x') : x.1 =[p] x'.1 :=
  begin induction q, constructor end

  definition is_prop_elimo_self {A : Type} (B : A → Type) {a : A} (b : B a) {H : is_prop (B a)} :
    @is_prop.elimo A B a a idp b b H = idpo :=
  !is_prop.elim

  definition sigma_pathover_equiv_of_is_prop {A : Type} {B : A → Type} (C : Πa, B a → Type)
    {a a' : A} (p : a = a') (x : Σb, C a b) (x' : Σb', C a' b')
    [Πa b, is_prop (C a b)] : x =[p] x' ≃ x.1 =[p] x'.1 :=
  begin
    fapply equiv.MK,
    { exact pathover_pr1 },
    { intro q, induction x with b c, induction x' with b' c', esimp at q, induction q,
      apply pathover_idp_of_eq, exact sigma_eq idp !is_prop.elimo },
    { intro q, induction x with b c, induction x' with b' c', esimp at q, induction q,
      have c = c', from !is_prop.elim, induction this,
      rewrite [▸*, is_prop_elimo_self (C a) c] },
    { intro q, induction q, induction x with b c, rewrite [▸*, is_prop_elimo_self (C a) c] }
  end

  definition sigma_ua {A B : Type} (C : A ≃ B → Type) :
    (Σ(p : A = B), C (equiv_of_eq p)) ≃ Σ(e : A ≃ B), C e :=
  sigma_equiv_sigma_left' !eq_equiv_equiv

  -- definition sigma_pathover_equiv_of_is_prop {A : Type} {B : A → Type} {C : Πa, B a → Type}
  --   {a a' : A} {p : a = a'} {b : B a} {b' : B a'} {c : C a b} {c' : C a' b'}
  --   [Πa b, is_prop (C a b)] : ⟨b, c⟩ =[p] ⟨b', c'⟩ ≃ b =[p] b' :=
  -- begin
  --   fapply equiv.MK,
  --   { exact pathover_pr1 },
  --   { intro q, induction q, apply pathover_idp_of_eq, exact sigma_eq idp !is_prop.elimo },
  --   { intro q, induction q,
  --     have c = c', from !is_prop.elim, induction this,
  --     rewrite [▸*, is_prop_elimo_self (C a) c] },
  --   { esimp, generalize ⟨b, c⟩, intro x q, }
  -- end
--rexact @(ap pathover_pr1) _ idpo _,

end sigma open sigma

namespace pointed

  definition phomotopy_of_homotopy {X Y : Type*} {f g : X →* Y} (h : f ~ g) [is_set Y] : f ~* g :=
  begin
    fapply phomotopy.mk,
    { exact h },
    { apply is_set.elim }
  end

end pointed open pointed

namespace group
  open is_trunc

  definition to_fun_isomorphism_trans {G H K : Group} (φ : G ≃g H) (ψ : H ≃g K) :
    φ ⬝g ψ ~ ψ ∘ φ :=
  by reflexivity

  definition add_homomorphism.mk [constructor] {G H : AddGroup} (φ : G → H) (h : is_add_hom φ) : G →g H :=
  homomorphism.mk φ h

  definition homomorphism_add [constructor] {G H : AddAbGroup} (φ ψ : G →g H) : G →g H :=
  add_homomorphism.mk (λg, φ g + ψ g)
    abstract begin
      intro g g', refine ap011 add !to_respect_add !to_respect_add ⬝ _,
      refine !add.assoc ⬝ ap (add _) (!add.assoc⁻¹ ⬝ ap (λx, x * _) !add.comm ⬝ !add.assoc) ⬝ !add.assoc⁻¹
    end end

  definition pmap_of_homomorphism_gid (G : Group) : pmap_of_homomorphism (gid G) ~* pid G :=
  begin
    fapply phomotopy_of_homotopy, reflexivity
  end

  definition pmap_of_homomorphism_gcompose {G H K : Group} (ψ : H →g K) (φ : G →g H)
    : pmap_of_homomorphism (ψ ∘g φ) ~* pmap_of_homomorphism ψ ∘* pmap_of_homomorphism φ :=
  begin
    fapply phomotopy_of_homotopy, reflexivity
  end

  definition pmap_of_homomorphism_phomotopy {G H : Group} {φ ψ : G →g H} (H : φ ~ ψ)
    : pmap_of_homomorphism φ ~* pmap_of_homomorphism ψ :=
  begin
    fapply phomotopy_of_homotopy, exact H
  end

  definition pequiv_of_isomorphism_trans {G₁ G₂ G₃ : Group} (φ : G₁ ≃g G₂) (ψ : G₂ ≃g G₂) :
    pequiv_of_isomorphism (φ ⬝g ψ) ~* pequiv_of_isomorphism ψ ∘* pequiv_of_isomorphism φ :=
  begin
    apply phomotopy_of_homotopy, reflexivity
  end

  definition isomorphism_eq {G H : Group} {φ ψ : G ≃g H} (p : φ ~ ψ) : φ = ψ :=
  begin
    induction φ with φ φe, induction ψ with ψ ψe,
    exact apd011 isomorphism.mk (homomorphism_eq p) !is_prop.elimo
  end

  definition is_set_isomorphism [instance] (G H : Group) : is_set (G ≃g H) :=
  begin
    have H : G ≃g H ≃ Σ(f : G →g H), is_equiv f,
    begin
      fapply equiv.MK,
      { intro φ, induction φ, constructor, assumption },
      { intro v, induction v, constructor, assumption },
      { intro v, induction v, reflexivity },
      { intro φ, induction φ, reflexivity }
    end,
    apply is_trunc_equiv_closed_rev, exact H
  end


--  definition is_equiv_isomorphism


  -- some extra instances for type class inference
  -- definition is_mul_hom_comm_homomorphism [instance] {G G' : AbGroup} (φ : G →g G')
  --   : @is_mul_hom G G' (@ab_group.to_group _ (AbGroup.struct G))
  --                           (@ab_group.to_group _ (AbGroup.struct G')) φ :=
  -- homomorphism.struct φ

  -- definition is_mul_hom_comm_homomorphism1 [instance] {G G' : AbGroup} (φ : G →g G')
  --   : @is_mul_hom G G' _
  --                           (@ab_group.to_group _ (AbGroup.struct G')) φ :=
  -- homomorphism.struct φ

  -- definition is_mul_hom_comm_homomorphism2 [instance] {G G' : AbGroup} (φ : G →g G')
  --   : @is_mul_hom G G' (@ab_group.to_group _ (AbGroup.struct G)) _ φ :=
  -- homomorphism.struct φ

end group open group

namespace fiber

  definition pcompose_ppoint {A B : Type*} (f : A →* B) : f ∘* ppoint f ~* pconst (pfiber f) B :=
  begin
    fapply phomotopy.mk,
    { exact point_eq },
    { exact !idp_con⁻¹ }
  end

  definition ap1_ppoint_phomotopy {A B : Type*} (f : A →* B)
    : Ω→ (ppoint f) ∘* pfiber_loop_space f ~* ppoint (Ω→ f) :=
  begin
    exact sorry
  end

  definition pfiber_equiv_of_square_ppoint {A B C D : Type*} {f : A →* B} {g : C →* D}
    (h : A ≃* C) (k : B ≃* D) (s : k ∘* f ~* g ∘* h)
    : ppoint g ∘* pfiber_equiv_of_square h k s ~* h ∘* ppoint f :=
  sorry

end fiber

namespace is_trunc

  definition center' {A : Type} (H : is_contr A) : A := center A

  definition pequiv_punit_of_is_contr [constructor] (A : Type*) (H : is_contr A) : A ≃* punit :=
  pequiv_of_equiv (equiv_unit_of_is_contr A) (@is_prop.elim unit _ _ _)

  definition pequiv_punit_of_is_contr' [constructor] (A : Type) (H : is_contr A)
    : pointed.MK A (center A) ≃* punit :=
  pequiv_punit_of_is_contr (pointed.MK A (center A)) H


definition is_trunc_is_contr_fiber [instance] [priority 900] (n : ℕ₋₂) {A B : Type} (f : A → B)
  (b : B) [is_trunc n A] [is_trunc n B] : is_trunc n (is_contr (fiber f b)) :=
begin
  cases n,
  { apply is_contr_of_inhabited_prop, apply is_contr_fun_of_is_equiv,
    apply is_equiv_of_is_contr },
  { apply is_trunc_succ_of_is_prop }
end

end is_trunc

namespace is_conn

  open unit trunc_index nat is_trunc pointed.ops

  definition is_contr_of_trivial_homotopy' (n : ℕ₋₂) (A : Type) [is_trunc n A] [is_conn -1 A]
    (H : Πk a, is_contr (π[k] (pointed.MK A a))) : is_contr A :=
  begin
    assert aa : trunc -1 A,
    { apply center },
    assert H3 : is_conn 0 A,
    { induction aa with a, exact H 0 a },
    exact is_contr_of_trivial_homotopy n A H
  end

  -- don't make is_prop_is_trunc an instance
  definition is_trunc_succ_is_trunc [instance] (n m : ℕ₋₂) (A : Type) : is_trunc (n.+1) (is_trunc m A) :=
  is_trunc_of_le _ !minus_one_le_succ

  definition is_conn_of_trivial_homotopy (n : ℕ₋₂) (m : ℕ) (A : Type) [is_trunc n A] [is_conn 0 A]
    (H : Π(k : ℕ) a, k ≤ m → is_contr (π[k] (pointed.MK A a))) : is_conn m A :=
  begin
    apply is_contr_of_trivial_homotopy_nat m (trunc m A),
    intro k a H2,
    induction a with a,
    apply is_trunc_equiv_closed_rev,
      exact equiv_of_pequiv (homotopy_group_trunc_of_le (pointed.MK A a) _ _ H2),
    exact H k a H2
  end

  definition is_conn_of_trivial_homotopy_pointed (n : ℕ₋₂) (m : ℕ) (A : Type*) [is_trunc n A]
    (H : Π(k : ℕ), k ≤ m → is_contr (π[k] A)) : is_conn m A :=
  begin
    have is_conn 0 A, proof H 0 !zero_le qed,
    apply is_conn_of_trivial_homotopy n m A,
    intro k a H2, revert a, apply is_conn.elim -1,
    cases A with A a, exact H k H2
  end

end is_conn

namespace circle

/-
  Suppose for `f, g : A -> B` I prove a homotopy `H : f ~ g` by induction on the element in `A`.
  And suppose `p : a = a'` is a path constructor in `A`.
  Then `natural_square_tr H p` has type `square (H a) (H a') (ap f p) (ap g p)` and is equal
  to the square which defined H on the path constructor
-/

  definition natural_square_elim_loop {A : Type} {f g : S¹ → A} (p : f base = g base)
    (q : square p p (ap f loop) (ap g loop))
    : natural_square (circle.rec p (eq_pathover q)) loop = q :=
  begin
    -- refine !natural_square_eq ⬝ _,
    refine ap square_of_pathover !rec_loop ⬝ _,
    exact to_right_inv !eq_pathover_equiv_square q
  end

  definition circle_elim_constant [unfold 5] {A : Type} {a : A} {p : a = a} (r : p = idp) (x : S¹) :
    circle.elim a p x = a :=
  begin
    induction x,
    { reflexivity },
    { apply eq_pathover_constant_right, apply hdeg_square, exact !elim_loop ⬝ r }
  end



end circle

namespace susp

  definition loop_psusp_intro_natural {X Y Z : Type*} (g : psusp Y →* Z) (f : X →* Y) :
    loop_psusp_intro (g ∘* psusp_functor f) ~* loop_psusp_intro g ∘* f :=
  pwhisker_right _ !ap1_pcompose ⬝* !passoc ⬝* pwhisker_left _ !loop_psusp_unit_natural⁻¹* ⬝*
  !passoc⁻¹*

  definition psusp_functor_phomotopy {A B : Type*} {f g : A →* B} (p : f ~* g) :
    psusp_functor f ~* psusp_functor g :=
  begin
    fapply phomotopy.mk,
    { intro x, induction x,
      { reflexivity },
      { reflexivity },
      { apply eq_pathover, apply hdeg_square, esimp, refine !elim_merid ⬝ _ ⬝ !elim_merid⁻¹ᵖ,
        exact ap merid (p a), }},
    { reflexivity },
  end

  definition psusp_functor_pid (A : Type*) : psusp_functor (pid A) ~* pid (psusp A) :=
  begin
    fapply phomotopy.mk,
    { intro x, induction x,
      { reflexivity },
      { reflexivity },
      { apply eq_pathover_id_right, apply hdeg_square, apply elim_merid }},
    { reflexivity },
  end

  definition psusp_functor_pcompose {A B C : Type*} (g : B →* C) (f : A →* B) :
    psusp_functor (g ∘* f) ~* psusp_functor g ∘* psusp_functor f :=
  begin
    fapply phomotopy.mk,
    { intro x, induction x,
      { reflexivity },
      { reflexivity },
      { apply eq_pathover, apply hdeg_square, esimp,
        refine !elim_merid ⬝ _ ⬝ (ap_compose (psusp_functor g) _ _)⁻¹ᵖ,
        refine _ ⬝ ap02 _ !elim_merid⁻¹, exact !elim_merid⁻¹ }},
    { reflexivity },
  end

  definition psusp_elim_psusp_functor {A B C : Type*} (g : B →* Ω C) (f : A →* B) :
    psusp.elim g ∘* psusp_functor f ~* psusp.elim (g ∘* f) :=
  begin
    refine !passoc ⬝* _, exact pwhisker_left _ !psusp_functor_pcompose⁻¹*
  end

  definition psusp_elim_phomotopy {A B : Type*} {f g : A →* Ω B} (p : f ~* g) : psusp.elim f ~* psusp.elim g :=
  pwhisker_left _ (psusp_functor_phomotopy p)

  definition psusp_elim_natural {X Y Z : Type*} (g : Y →* Z) (f : X →* Ω Y)
    : g ∘* psusp.elim f ~* psusp.elim (Ω→ g ∘* f) :=
  begin
    refine _ ⬝* pwhisker_left _ !psusp_functor_pcompose⁻¹*,
    refine !passoc⁻¹* ⬝* _ ⬝* !passoc,
    exact pwhisker_right _ !loop_psusp_counit_natural
  end

end susp

namespace category

  -- replace precategory_group with precategory_Group (the former has a universe error)
  definition precategory_Group.{u} [instance] [constructor] : precategory.{u+1 u} Group :=
  begin
    fapply precategory.mk,
    { exact λG H, G →g H },
    { exact _ },
    { exact λG H K ψ φ, ψ ∘g φ },
    { exact λG, gid G },
    { intros, apply homomorphism_eq, esimp },
    { intros, apply homomorphism_eq, esimp },
    { intros, apply homomorphism_eq, esimp }
  end


  definition precategory_AbGroup.{u} [instance] [constructor] : precategory.{u+1 u} AbGroup :=
  begin
    fapply precategory.mk,
    { exact λG H, G →g H },
    { exact _ },
    { exact λG H K ψ φ, ψ ∘g φ },
    { exact λG, gid G },
    { intros, apply homomorphism_eq, esimp },
    { intros, apply homomorphism_eq, esimp },
    { intros, apply homomorphism_eq, esimp }
  end
  open iso
  definition Group_is_iso_of_is_equiv {G H : Group} (φ : G →g H) (H : is_equiv (group_fun φ)) :
    is_iso φ :=
  begin
    fconstructor,
    { exact (isomorphism.mk φ H)⁻¹ᵍ },
    { apply homomorphism_eq, rexact left_inv φ },
    { apply homomorphism_eq, rexact right_inv φ }
  end

  definition Group_is_equiv_of_is_iso {G H : Group} (φ : G ⟶ H) (Hφ : is_iso φ) :
    is_equiv (group_fun φ) :=
  begin
    fapply adjointify,
    { exact group_fun φ⁻¹ʰ },
    { note p := right_inverse φ, exact ap010 group_fun p },
    { note p := left_inverse φ,  exact ap010 group_fun p }
  end

  definition Group_iso_equiv (G H : Group) : (G ≅ H) ≃ (G ≃g H) :=
  begin
    fapply equiv.MK,
    { intro φ, induction φ with φ φi, constructor, exact Group_is_equiv_of_is_iso φ _ },
    { intro v, induction v with φ φe, constructor, exact Group_is_iso_of_is_equiv φ _ },
    { intro v, induction v with φ φe, apply isomorphism_eq, reflexivity },
    { intro φ, induction φ with φ φi, apply iso_eq, reflexivity }
  end

  definition Group_props.{u} {A : Type.{u}} (v : (A → A → A) × (A → A) × A) : Prop.{u} :=
  begin
    induction v with m v, induction v with i o,
    fapply trunctype.mk,
    { exact is_set A × (Πa, m a o = a) × (Πa, m o a = a) × (Πa b c, m (m a b) c = m a (m b c)) ×
      (Πa, m (i a) a = o) },
    { apply is_trunc_of_imp_is_trunc, intro v, induction v with H v,
      have is_prop (Πa, m a o = a), from _,
      have is_prop (Πa, m o a = a), from _,
      have is_prop (Πa b c, m (m a b) c = m a (m b c)), from _,
      have is_prop (Πa, m (i a) a = o), from _,
      apply is_trunc_prod }
  end

  definition Group.sigma_char2.{u} : Group.{u} ≃
    Σ(A : Type.{u}) (v : (A → A → A) × (A → A) × A), Group_props v :=
  begin
    fapply equiv.MK,
    { intro G, refine ⟨G, _⟩, induction G with G g, induction g with m s ma o om mo i mi,
      repeat (fconstructor; do 2 try assumption), },
    { intro v, induction v with x v, induction v with y v, repeat induction y with x y,
      repeat induction v with x v, constructor, fconstructor, repeat assumption },
    { intro v, induction v with x v, induction v with y v, repeat induction y with x y,
      repeat induction v with x v, reflexivity },
    { intro v, repeat induction v with x v, reflexivity },
  end
  open is_trunc

  section
  local attribute group.to_has_mul group.to_has_inv [coercion]

  theorem inv_eq_of_mul_eq {A : Type} (G H : group A) (p : @mul A G ~2 @mul A H) :
    @inv A G ~ @inv A H :=
  begin
    have foo : Π(g : A), @inv A G g = (@inv A G g * g) * @inv A H g,
      from λg, !mul_inv_cancel_right⁻¹,
    cases G with Gs Gm Gh1 G1 Gh2 Gh3 Gi Gh4,
    cases H with Hs Hm Hh1 H1 Hh2 Hh3 Hi Hh4,
    change Gi ~ Hi, intro g, have p' : Gm ~2 Hm, from p,
    calc
      Gi g = Hm (Hm (Gi g) g) (Hi g) : foo
       ... = Hm (Gm (Gi g) g) (Hi g) : by rewrite p'
       ... = Hm G1 (Hi g) : by rewrite Gh4
       ... = Gm G1 (Hi g) : by rewrite p'
       ... = Hi g : Gh2
  end

  theorem one_eq_of_mul_eq {A : Type} (G H : group A)
    (p : @mul A (group.to_has_mul G) ~2 @mul A (group.to_has_mul H)) :
    @one A (group.to_has_one G) = @one A (group.to_has_one H) :=
  begin
    cases G with Gm Gs Gh1 G1 Gh2 Gh3 Gi Gh4,
    cases H with Hm Hs Hh1 H1 Hh2 Hh3 Hi Hh4,
    exact (Hh2 G1)⁻¹ ⬝ (p H1 G1)⁻¹ ⬝ Gh3 H1,
  end
  end

  open prod.ops
  definition group_of_Group_props.{u} {A : Type.{u}} {m : A → A → A} {i : A → A} {o : A}
    (H : Group_props (m, (i, o))) : group A :=
  ⦃group, mul := m, inv := i, one := o, is_set_carrier := H.1,
    mul_one := H.2.1, one_mul := H.2.2.1, mul_assoc := H.2.2.2.1, mul_left_inv := H.2.2.2.2⦄

  theorem Group_eq_equiv_lemma2 {A : Type} {m m' : A → A → A} {i i' : A → A} {o o' : A}
    (H : Group_props (m, (i, o))) (H' : Group_props (m', (i', o'))) :
    (m, (i, o)) = (m', (i', o')) ≃ (m ~2 m') :=
  begin
    have is_set A, from pr1 H,
    apply equiv_of_is_prop,
    { intro p, exact apd100 (eq_pr1 p)},
    { intro p, apply prod_eq (eq_of_homotopy2 p),
      apply prod_eq: esimp [Group_props] at *; esimp,
      { apply eq_of_homotopy,
        exact inv_eq_of_mul_eq (group_of_Group_props H) (group_of_Group_props H') p },
      { exact one_eq_of_mul_eq (group_of_Group_props H) (group_of_Group_props H') p }}
  end

  open sigma.ops

  theorem Group_eq_equiv_lemma {G H : Group}
    (p : (Group.sigma_char2 G).1 = (Group.sigma_char2 H).1) :
    ((Group.sigma_char2 G).2 =[p] (Group.sigma_char2 H).2) ≃
    (is_mul_hom (equiv_of_eq (proof p qed : Group.carrier G = Group.carrier H))) :=
  begin
    refine !sigma_pathover_equiv_of_is_prop ⬝e _,
    induction G with G g, induction H with H h,
    esimp [Group.sigma_char2] at p, induction p,
    refine !pathover_idp ⬝e _,
    induction g with s m ma o om mo i mi, induction h with σ μ μa ε εμ με ι μι,
    exact Group_eq_equiv_lemma2 (Group.sigma_char2 (Group.mk G (group.mk s m ma o om mo i mi))).2.2
                                (Group.sigma_char2 (Group.mk G (group.mk σ μ μa ε εμ με ι μι))).2.2
  end

  definition isomorphism.sigma_char (G H : Group) : (G ≃g H) ≃ Σ(e : G ≃ H), is_mul_hom e :=
  begin
    fapply equiv.MK,
    { intro φ, exact ⟨equiv_of_isomorphism φ, to_respect_mul φ⟩ },
    { intro v, induction v with e p, exact isomorphism_of_equiv e p },
    { intro v, induction v with e p, induction e, reflexivity },
    { intro φ, induction φ with φ H, induction φ, reflexivity },
  end

  definition Group_eq_equiv (G H : Group) : G = H ≃ (G ≃g H) :=
  begin
    refine (eq_equiv_fn_eq_of_equiv Group.sigma_char2 G H) ⬝e _,
    refine !sigma_eq_equiv ⬝e _,
    refine sigma_equiv_sigma_right Group_eq_equiv_lemma ⬝e _,
    transitivity (Σ(e : (Group.sigma_char2 G).1 ≃ (Group.sigma_char2 H).1),
      @is_mul_hom _ _ _ _ (to_fun e)), apply sigma_ua,
    exact !isomorphism.sigma_char⁻¹ᵉ
  end

  definition to_fun_Group_eq_equiv {G H : Group} (p : G = H)
    : Group_eq_equiv G H p ~ isomorphism_of_eq p :=
  begin
    induction p, reflexivity
  end

  definition Group_eq2 {G H : Group} {p q : G = H}
    (r : isomorphism_of_eq p ~ isomorphism_of_eq q) : p = q :=
  begin
    apply eq_of_fn_eq_fn (Group_eq_equiv G H),
    apply isomorphism_eq,
    intro g, refine to_fun_Group_eq_equiv p g ⬝ r g ⬝ (to_fun_Group_eq_equiv q g)⁻¹,
  end

  definition Group_eq_equiv_Group_iso (G₁ G₂ : Group) : G₁ = G₂ ≃ G₁ ≅ G₂ :=
  Group_eq_equiv G₁ G₂ ⬝e (Group_iso_equiv G₁ G₂)⁻¹ᵉ

  definition category_Group.{u} : category Group.{u} :=
  category.mk precategory_Group
  begin
    intro G H,
    apply is_equiv_of_equiv_of_homotopy (Group_eq_equiv_Group_iso G H),
    intro p, induction p, fapply iso_eq, apply homomorphism_eq, reflexivity
  end

  definition category_AbGroup : category AbGroup :=
  category.mk precategory_AbGroup sorry

  definition Grp.{u} [constructor] : Category := category.Mk Group.{u} category_Group
  definition AbGrp [constructor] : Category := category.Mk AbGroup category_AbGroup

end category

namespace sphere

  definition psphere_pequiv_iterate_psusp (n : ℕ) : psphere n ≃* iterate_psusp n pbool :=
  begin
    induction n with n e,
    { exact psphere_pequiv_pbool },
    { exact psusp_pequiv e }
  end


  -- definition constant_sphere_map_sphere {n m : ℕ} (H : n < m) (f : S* n →* S* m) :
  --   f ~* pconst (S* n) (S* m) :=
  -- begin
  --   assert H : is_contr (Ω[n] (S* m)),
  --   { apply homotopy_group_sphere_le, },
  --   apply phomotopy_of_eq,
  --   apply eq_of_fn_eq_fn !psphere_pmap_pequiv,
  --   apply @is_prop.elim
  -- end

end sphere

definition image_pathover {A B : Type} (f : A → B) {x y : B} (p : x = y) (u : image f x) (v : image f y) : u =[p] v :=
  begin
    apply is_prop.elimo
  end

section injective_surjective
open trunc fiber image

variables {A B C : Type} [is_set A] [is_set B] [is_set C] (f : A → B) (g : B → C) (h : A → C) (H : g ∘ f ~ h)
include H

definition is_embedding_factor : is_embedding h → is_embedding f :=
  begin
    induction H using homotopy.rec_on_idp,
    intro E,
    fapply is_embedding_of_is_injective,
    intro x y p,
    fapply @is_injective_of_is_embedding _ _ _ E _ _ (ap g p)
  end

definition is_surjective_factor : is_surjective h → is_surjective g :=
  begin
    induction H using homotopy.rec_on_idp,
    intro S,
    intro c,
    note p := S c,
    induction p,
    apply tr,
    fapply fiber.mk,
    exact f a,
    exact p
  end

end injective_surjective

definition AbGroup_of_Group.{u} (G : Group.{u}) (H : Π x y : G, x * y = y * x) : AbGroup.{u} :=
begin
  induction G,
  fapply AbGroup.mk,
  assumption,
  exact ⦃ab_group, struct, mul_comm := H⦄
end

definition trivial_ab_group : AbGroup.{0} :=
begin
  fapply AbGroup_of_Group Trivial_group, intro x y, reflexivity
end

definition trivial_homomorphism (A B : AbGroup) : A →g B :=
begin
  fapply homomorphism.mk,
  exact λ a, 1,
  intros, symmetry, exact one_mul 1,
end

definition from_trivial_ab_group (A : AbGroup) :  trivial_ab_group →g A :=
  trivial_homomorphism trivial_ab_group A

definition is_embedding_from_trivial_ab_group (A : AbGroup) : is_embedding (from_trivial_ab_group A) :=
  begin
    fapply is_embedding_of_is_injective,
    intro x y p,
    induction x, induction y, reflexivity
  end

definition to_trivial_ab_group (A : AbGroup) : A →g trivial_ab_group :=
  trivial_homomorphism A trivial_ab_group

/- Stuff added by Jeremy -/

definition exists.elim {A : Type} {p : A → Type} {B : Type} [is_prop B] (H : Exists p)
  (H' : ∀ (a : A), p a → B) : B :=
trunc.elim (sigma.rec H') H

definition image.elim {A B : Type} {f : A → B} {C : Type} [is_prop C] {b : B}
  (H : image f b) (H' : ∀ (a : A), f a = b → C) : C :=
begin
  refine (trunc.elim _ H),
  intro H'', cases H'' with a Ha, exact H' a Ha
end

definition image.intro {A B : Type} {f : A → B} {a : A} {b : B} (h : f a = b) : image f b :=
begin
  apply trunc.merely.intro,
  apply fiber.mk,
  exact h
end

definition image.equiv_exists {A B : Type} {f : A → B} {b : B} : image f b ≃ ∃ a, f a = b :=
trunc_equiv_trunc _ (fiber.sigma_char _ _)

-- move to homomorphism.hlean
section
  theorem eq_zero_of_eq_zero_of_is_embedding {A B : Type} [add_group A] [add_group B]
    {f : A → B} [is_add_hom f] [is_embedding f] {a : A} (h : f a = 0) : a = 0 :=
  have f a = f 0, by rewrite [h, respect_zero],
  show a = 0, from is_injective_of_is_embedding this
end

/- put somewhere in algebra -/

structure Ring :=
(carrier : Type) (struct : ring carrier)

attribute Ring.carrier [coercion]
attribute Ring.struct [instance]