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

import homotopy.sphere2 homotopy.cofiber homotopy.wedge hit.prop_trunc hit.set_quotient

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

attribute is_prop.elim_set [unfold 6]

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

-- move to chain_complex (or another file). rename chain_complex.is_exact
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

definition is_exact.im_in_ker2 {A B : Type} {C : Set*} {f : A → B} {g : B → C} (H : is_exact f g)
  {b : B} (h : image f b) : g b = pt :=
begin
  induction h with a p, exact ap g p⁻¹ ⬝ is_exact.im_in_ker H a
end

-- TO DO: give less univalency proof
definition is_exact_homotopy {A B : Type} {C : Type*} {f f' : A → B} {g g' : B → C}
  (p : f ~ f') (q : g ~ g') (H : is_exact f g) : is_exact f' g' :=
begin
  induction p using homotopy.rec_on_idp,
  induction q using homotopy.rec_on_idp,
  exact H
end

definition is_exact_trunc_functor {A B : Type} {C : Type*} {f : A → B} {g : B → C}
  (H : is_exact_t f g) : @is_exact _ _ (ptrunc 0 C) (trunc_functor 0 f) (trunc_functor 0 g) :=
begin
  constructor,
  { intro a, esimp, induction a with a,
    exact ap tr (is_exact_t.im_in_ker H a) },
  { intro b p, induction b with b, note q := !tr_eq_tr_equiv p, induction q with q,
    induction is_exact_t.ker_in_im H b q with a r,
    exact image.mk (tr a) (ap tr r) }
end

definition is_contr_middle_of_is_exact {A B : Type} {C : Type*} {f : A → B} {g : B → C} (H : is_exact f g)
  [is_contr A] [is_set B] [is_contr C] : is_contr B :=
begin
  apply is_contr.mk (f pt),
  intro b,
  induction is_exact.ker_in_im H b !is_prop.elim,
  exact ap f !is_prop.elim ⬝ p
end

definition is_surjective_of_is_exact_of_is_contr {A B : Type} {C : Type*} {f : A → B} {g : B → C}
  (H : is_exact f g) [is_contr C] : is_surjective f :=
λb, is_exact.ker_in_im H b !is_prop.elim

section chain_complex
open succ_str chain_complex
definition is_exact_of_is_exact_at {N : succ_str} {A : chain_complex N} {n : N}
  (H : is_exact_at A n) : is_exact (cc_to_fn A (S n)) (cc_to_fn A n) :=
is_exact.mk (cc_is_chain_complex A n) H
end chain_complex

namespace algebra
  definition ab_group_unit [constructor] : ab_group unit :=
  ⦃ab_group, trivial_group, mul_comm := λx y, idp⦄

  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

  definition pSet_of_AddGroup [constructor] [reducible] [coercion] (G : AddGroup) : Set* :=
  pSet_of_Group G
  attribute algebra._trans_of_pSet_of_AddGroup [unfold 1]
  attribute algebra._trans_of_pSet_of_AddGroup_1 algebra._trans_of_pSet_of_AddGroup_2 [constructor]

  definition pType_of_AddGroup [reducible] [constructor] : AddGroup → Type* :=
  algebra._trans_of_pSet_of_AddGroup_1
  definition Set_of_AddGroup [reducible] [constructor] : AddGroup → Set :=
  algebra._trans_of_pSet_of_AddGroup_2

  -- --
  -- definition Group_of_AddAbGroup [coercion] [constructor] (G : AddAbGroup) : Group :=
  -- AddGroup.mk G _
  -- --

  definition AddGroup_of_AddAbGroup [coercion] [constructor] (G : AddAbGroup) : AddGroup :=
  AddGroup.mk G _

  attribute algebra._trans_of_AddGroup_of_AddAbGroup_1
            algebra._trans_of_AddGroup_of_AddAbGroup
            algebra._trans_of_AddGroup_of_AddAbGroup_3 [constructor]
  attribute algebra._trans_of_AddGroup_of_AddAbGroup_2 [unfold 1]

  definition add_ab_group_AddAbGroup2 [instance] (G : AddAbGroup) : add_ab_group G :=
  AddAbGroup.struct G

end algebra

namespace eq

  definition eq.rec_to {A : Type} {a₀ : A} {P : Π⦃a₁⦄, a₀ = a₁ → Type}
    {a₁ : A} (p₀ : a₀ = a₁) (H : P p₀) ⦃a₂ : A⦄ (p : a₀ = a₂) : P p :=
  begin
    induction p₀, induction p, exact H
  end

  definition eq.rec_to2 {A : Type} {P : Π⦃a₀ a₁⦄, a₀ = a₁ → Type}
    {a₀ a₀' a₁' : A} (p' : a₀' = a₁') (p₀ : a₀ = a₀') (H : P p') ⦃a₁ : A⦄ (p : a₀ = a₁) : P p :=
  begin
   induction p₀, induction p', induction p, exact H
  end

  definition eq.rec_right_inv {A : Type} (f : A ≃ A) {P : Π⦃a₀ a₁⦄, f a₀ = a₁ → Type}
    (H : Πa, P (right_inv f a)) ⦃a₀ a₁ : A⦄ (p : f a₀ = a₁) : P p :=
  begin
    revert a₀ p, refine equiv_rect f⁻¹ᵉ _ _,
    intro a₀ p, exact eq.rec_to (right_inv f a₀) (H a₀) p,
  end

  definition eq.rec_equiv {A B : Type} {a₀ : A} (f : A ≃ B) {P : Π{a₁}, f a₀ = f a₁ → Type}
    (H : P (idpath (f a₀))) ⦃a₁ : A⦄ (p : f a₀ = f a₁) : P p :=
  begin
    assert qr : Σ(q : a₀ = a₁), ap f q = p,
    { exact ⟨eq_of_fn_eq_fn f p, ap_eq_of_fn_eq_fn' f p⟩ },
    cases qr with q r, apply transport P r, induction q, exact H
  end

  definition eq.rec_equiv_symm {A B : Type} {a₁ : A} (f : A ≃ B) {P : Π{a₀}, f a₀ = f a₁ → Type}
    (H : P (idpath (f a₁))) ⦃a₀ : A⦄ (p : f a₀ = f a₁) : P p :=
  begin
    assert qr : Σ(q : a₀ = a₁), ap f q = p,
    { exact ⟨eq_of_fn_eq_fn f p, ap_eq_of_fn_eq_fn' f p⟩ },
    cases qr with q r, apply transport P r, induction q, exact H
  end

  definition eq.rec_equiv_to_same {A B : Type} {a₀ : A} (f : A ≃ B) {P : Π{a₁}, f a₀ = f a₁ → Type}
    ⦃a₁' : A⦄ (p' : f a₀ = f a₁') (H : P p') ⦃a₁ : A⦄ (p : f a₀ = f a₁) : P p :=
  begin
    revert a₁' p' H a₁ p,
    refine eq.rec_equiv f _,
    exact eq.rec_equiv f
  end

  definition eq.rec_equiv_to {A A' B : Type} {a₀ : A} (f : A ≃ B) (g : A' ≃ B)
    {P : Π{a₁}, f a₀ = g a₁ → Type}
    ⦃a₁' : A'⦄ (p' : f a₀ = g a₁') (H : P p') ⦃a₁ : A'⦄ (p : f a₀ = g a₁) : P p :=
  begin
    assert qr : Σ(q : g⁻¹ (f a₀) = a₁), (right_inv g (f a₀))⁻¹ ⬝ ap g q = p,
    { exact ⟨eq_of_fn_eq_fn g (right_inv g (f a₀) ⬝ p),
             whisker_left _ (ap_eq_of_fn_eq_fn' g _) ⬝ !inv_con_cancel_left⟩ },
    assert q'r' : Σ(q' : g⁻¹ (f a₀) = a₁'), (right_inv g (f a₀))⁻¹ ⬝ ap g q' = p',
    { exact ⟨eq_of_fn_eq_fn g (right_inv g (f a₀) ⬝ p'),
             whisker_left _ (ap_eq_of_fn_eq_fn' g _) ⬝ !inv_con_cancel_left⟩ },
    induction qr with q r, induction q'r' with q' r',
    induction q, induction q',
    induction r, induction r',
    exact H
  end

  definition eq.rec_grading {A A' B : Type} {a : A} (f : A ≃ B) (g : A' ≃ B)
    {P : Π{b}, f a = b → Type}
    {a' : A'} (p' : f a = g a') (H : P p') ⦃b : B⦄ (p : f a = b) : P p :=
  begin
    revert b p, refine equiv_rect g _ _,
    exact eq.rec_equiv_to f g p' H
  end

  definition eq.rec_grading_unbased {A B B' C : Type} (f : A ≃ B) (g : B ≃ C) (h : B' ≃ C)
    {P : Π{b c}, g b = c → Type}
    {a' : A} {b' : B'} (p' : g (f a') = h b') (H : P p') ⦃b : B⦄ ⦃c : C⦄ (q : f a' = b)
    (p : g b = c) : P p :=
  begin
    induction q, exact eq.rec_grading (f ⬝e g) h p' H p
  end

  definition eq.rec_symm {A : Type} {a₀ : A} {P : Π⦃a₁⦄, a₁ = a₀ → Type}
    (H : P idp) ⦃a₁ : A⦄ (p : a₁ = a₀) : P p :=
  begin
    cases p, exact H
  end

  definition is_contr_homotopy_group_of_is_contr (A : Type*) (n : ℕ) [is_contr A] : is_contr (π[n] A) :=
  begin
    apply is_trunc_trunc_of_is_trunc,
    apply is_contr_loop_of_is_trunc,
    apply is_trunc_of_is_contr
  end

  definition cast_fn_cast_square {A : Type} {B C : A → Type} (f : Π⦃a⦄, B a → C a) {a₁ a₂ : A}
    (p : a₁ = a₂) (q : a₂ = a₁) (r : p ⬝ q = idp) (b : B a₁) :
    cast (ap C q) (f (cast (ap B p) b)) = f b :=
  have q⁻¹ = p, from inv_eq_of_idp_eq_con r⁻¹,
  begin induction this, induction q, reflexivity end

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

  definition inv_homotopy_inv {A B : Type} {f g : A → B} [is_equiv f] [is_equiv g]
    (p : f ~ g) : f⁻¹ ~ g⁻¹ :=
  λa, inv_eq_of_eq (p (g⁻¹ a) ⬝ right_inv g a)⁻¹

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

  definition compose2 {A B C : Type} {g g' : B → C} {f f' : A → B}
    (p : g ~ g') (q : f ~ f') : g ∘ f ~ g' ∘ f' :=
  hwhisker_right f p ⬝hty hwhisker_left g' q

  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 homotopy_top_of_hsquare {f₂₁ : A₂₀ ≃ A₂₂} (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) :
    f₁₀ ~ f₂₁⁻¹ ∘ f₁₂ ∘ f₀₁ :=
  homotopy_inv_of_homotopy_post _ _ _ p

  definition homotopy_top_of_hsquare' [is_equiv f₂₁] (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) :
    f₁₀ ~ f₂₁⁻¹ ∘ f₁₂ ∘ f₀₁ :=
  homotopy_inv_of_homotopy_post _ _ _ 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 nat

  definition iterate_succ {A : Type} (f : A → A) (n : ℕ) (x : A) :
    f^[succ n] x = f^[n] (f x) :=
  by induction n with n p; reflexivity; exact ap f p

  lemma iterate_sub {A : Type} (f : A ≃ A) {n m : ℕ} (h : n ≥ m) (a : A) :
    iterate f (n - m) a = iterate f n (iterate f⁻¹ m a) :=
  begin
    revert n h, induction m with m p: intro n h,
    { reflexivity },
    { cases n with n, exfalso, apply not_succ_le_zero _ h,
      rewrite [succ_sub_succ], refine p n (le_of_succ_le_succ h) ⬝ _,
      refine ap (f^[n]) _ ⬝ !iterate_succ⁻¹, exact !to_right_inv⁻¹ }
  end

  definition iterate_commute {A : Type} {f g : A → A} (n : ℕ) (h : f ∘ g ~ g ∘ f) :
    iterate f n ∘ g ~ g ∘ iterate f n :=
  by induction n with n IH; reflexivity; exact λx, ap f (IH x) ⬝ !h

  definition iterate_equiv {A : Type} (f : A ≃ A) (n : ℕ) : A ≃ A :=
  equiv.mk (iterate f n)
           (by induction n with n IH; apply is_equiv_id; exact is_equiv_compose f (iterate f n))

  definition iterate_inv {A : Type} (f : A ≃ A) (n : ℕ) :
    (iterate_equiv f n)⁻¹ ~ iterate f⁻¹ n :=
  begin
    induction n with n p: intro a,
      reflexivity,
      exact p (f⁻¹ a) ⬝ !iterate_succ⁻¹
  end

  definition iterate_left_inv {A : Type} (f : A ≃ A) (n : ℕ) (a : A) : f⁻¹ᵉ^[n] (f^[n] a) = a :=
  (iterate_inv f n (f^[n] a))⁻¹ ⬝ to_left_inv (iterate_equiv f n) a

  definition iterate_right_inv {A : Type} (f : A ≃ A) (n : ℕ) (a : A) : f^[n] (f⁻¹ᵉ^[n] a) = a :=
  ap (f^[n]) (iterate_inv f n a)⁻¹ ⬝ to_right_inv (iterate_equiv f n) a

end nat

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 algebra

  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 (G H : AddGroup) : Type := homomorphism G H
  infix ` →a `:55 := add_homomorphism

  definition agroup_fun [coercion] [unfold 3] [reducible] {G H : AddGroup} (φ : G →a H) : G → H :=
  φ

  definition add_homomorphism.struct [instance] {G H : AddGroup} (φ : G →a H) : is_add_hom φ :=
  homomorphism.addstruct φ

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

  definition add_homomorphism_compose [constructor] [trans] {G₁ G₂ G₃ : AddGroup}
    (ψ : G₂ →a G₃) (φ : G₁ →a G₂) : G₁ →a G₃ :=
  add_homomorphism.mk (ψ ∘ φ) (is_add_hom_compose _ _)

  definition add_homomorphism_id [constructor] [refl] (G : AddGroup) : G →a G :=
  add_homomorphism.mk (@id G) (is_add_hom_id G)

  abbreviation aid [constructor] := @add_homomorphism_id
  infixr ` ∘a `:75 := add_homomorphism_compose

  definition to_respect_add' {H₁ H₂ : AddGroup} (χ : H₁ →a H₂) (g h : H₁) : χ (g + h) = χ g + χ h :=
  respect_add χ g h

  theorem to_respect_zero' {H₁ H₂ : AddGroup} (χ : H₁ →a H₂) : χ 0 = 0 :=
  respect_zero χ

  theorem to_respect_neg' {H₁ H₂ : AddGroup} (χ : H₁ →a H₂) (g : H₁) : χ (-g) = -(χ g) :=
  respect_neg χ g

  definition homomorphism_add [constructor] {G H : AddAbGroup} (φ ψ : G →a H) : G →a 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 homomorphism_mul [constructor] {G H : AbGroup} (φ ψ : G →g H) : G →g H :=
  homomorphism.mk (λg, φ g * ψ g) (to_respect_add (homomorphism_add φ ψ))

  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_mul_right [constructor] {A : Group} (a : A) : is_equiv (λb, b * a) :=
  adjointify _ (λb : A, b * a⁻¹) (λb, !inv_mul_cancel_right) (λb, !mul_inv_cancel_right)

  definition right_action [constructor] {A : Group} (a : A) : A ≃ A :=
  equiv.mk _ (is_equiv_mul_right a)

  definition is_equiv_add_right [constructor] {A : AddGroup} (a : A) : is_equiv (λb, b + a) :=
  adjointify _ (λb : A, b - a) (λb, !neg_add_cancel_right) (λb, !add_neg_cancel_right)

  definition add_right_action [constructor] {A : AddGroup} (a : A) : A ≃ A :=
  equiv.mk _ (is_equiv_add_right a)

  section
    variables {A B : Type} (f : A ≃ B) [ab_group A]
    definition group_equiv_mul_comm (b b' : B) : group_equiv_mul f b b' = group_equiv_mul f b' b :=
    by rewrite [↑group_equiv_mul, mul.comm]

    definition ab_group_equiv_closed : ab_group B :=
    ⦃ab_group, group_equiv_closed f,
      mul_comm := group_equiv_mul_comm f⦄
  end

  definition ab_group_of_is_contr (A : Type) [is_contr A] : ab_group A :=
  have ab_group unit, from ab_group_unit,
  ab_group_equiv_closed (equiv_unit_of_is_contr A)⁻¹ᵉ

  definition group_of_is_contr (A : Type) [is_contr A] : group A :=
  have ab_group A, from ab_group_of_is_contr A, by apply _

  definition ab_group_lift_unit : ab_group (lift unit) :=
  ab_group_of_is_contr (lift unit)

  definition trivial_ab_group_lift : AbGroup :=
  AbGroup.mk _ ab_group_lift_unit

  definition homomorphism_of_is_contr_right (A : Group) {B : Type} (H : is_contr B) :
    A →g Group.mk B (group_of_is_contr B) :=
  group.homomorphism.mk (λa, center _) (λa a', !is_prop.elim)

  open trunc pointed is_conn
  definition ab_group_homotopy_group_of_is_conn (n : ℕ) (A : Type*) [H : is_conn 1 A] :
    ab_group (π[n] A) :=
  begin
    have is_conn 0 A, from !is_conn_of_is_conn_succ,
    cases n with n,
    { unfold [homotopy_group, ptrunc], apply ab_group_of_is_contr },
    cases n with n,
    { unfold [homotopy_group, ptrunc], apply ab_group_of_is_contr },
    exact ab_group_homotopy_group n A
  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 function
  variables {A B : Type} {f f' : A → B}
  definition is_embedding_homotopy_closed (p : f ~ f') (H : is_embedding f) : is_embedding f' :=
  begin
    intro a a', fapply is_equiv_of_equiv_of_homotopy,
    exact equiv.mk (ap f) _ ⬝e equiv_eq_closed_left _ (p a) ⬝e equiv_eq_closed_right _ (p a'),
    intro q, esimp, exact (eq_bot_of_square (transpose (natural_square p q)))⁻¹
  end

  definition is_embedding_homotopy_closed_rev (p : f' ~ f) (H : is_embedding f) : is_embedding f' :=
  is_embedding_homotopy_closed p⁻¹ʰᵗʸ H

  definition is_surjective_homotopy_closed (p : f ~ f') (H : is_surjective f) : is_surjective f' :=
  begin
    intro b, induction H b with a q,
    exact image.mk a ((p a)⁻¹ ⬝ q)
  end

  definition is_surjective_homotopy_closed_rev (p : f' ~ f) (H : is_surjective f) :
    is_surjective f' :=
  is_surjective_homotopy_closed p⁻¹ʰᵗʸ H

  definition is_equiv_ap1_gen_of_is_embedding {A B : Type} (f : A → B) [is_embedding f]
    {a a' : A} {b b' : B} (q : f a = b) (q' : f a' = b') : is_equiv (ap1_gen f q q') :=
  begin
    induction q, induction q',
    exact is_equiv.homotopy_closed _ (ap1_gen_idp_left f)⁻¹ʰᵗʸ,
  end

  definition is_equiv_ap1_of_is_embedding {A B : Type*} (f : A →* B) [is_embedding f] :
    is_equiv (Ω→ f) :=
  is_equiv_ap1_gen_of_is_embedding f (respect_pt f) (respect_pt f)

  definition loop_pequiv_loop_of_is_embedding [constructor] {A B : Type*} (f : A →* B)
    [is_embedding f] : Ω A ≃* Ω B :=
  pequiv_of_pmap (Ω→ f) (is_equiv_ap1_of_is_embedding f)

  definition loopn_pequiv_loopn_of_is_embedding [constructor] (n : ℕ) [H : is_succ n]
    {A B : Type*} (f : A →* B) [is_embedding f] : Ω[n] A ≃* Ω[n] B :=
  begin
    induction H with n,
    exact !loopn_succ_in ⬝e*
      loopn_pequiv_loopn n (loop_pequiv_loop_of_is_embedding f) ⬝e*
      !loopn_succ_in⁻¹ᵉ*
  end

  definition homotopy_group_isomorphism_of_is_embedding (n : ℕ) [H : is_succ n] {A B : Type*}
    (f : A →* B) [H2 : is_embedding f] : πg[n] A ≃g πg[n] B :=
  begin
    apply isomorphism.mk (homotopy_group_homomorphism n f),
    induction H with n,
    apply is_equiv_of_equiv_of_homotopy
      (ptrunc_pequiv_ptrunc 0 (loopn_pequiv_loopn_of_is_embedding (n+1) f)),
    exact sorry
  end

end function open function

namespace fiber
  open pointed

  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 point_fiber_eq {A B : Type} {f : A → B} {b : B} {x y : fiber f b}
    (p : point x = point y) (q : point_eq x = ap f p ⬝ point_eq y) :
    ap point (fiber_eq p q) = p :=
  begin
    induction x with a r, induction y with a' s, esimp at *, induction p,
    induction q using eq.rec_symm, induction s, reflexivity
  end

  definition fiber_eq_equiv_fiber {A B : Type} {f : A → B} {b : B} (x y : fiber f b) :
    x = y ≃ fiber (ap1_gen f (point_eq x) (point_eq y)) (idpath b) :=
  calc
    x = y ≃ fiber.sigma_char f b x = fiber.sigma_char f b y :
      eq_equiv_fn_eq_of_equiv (fiber.sigma_char f b) x y
      ... ≃ Σ(p : point x = point y), point_eq x =[p] point_eq y : sigma_eq_equiv
      ... ≃ Σ(p : point x = point y), (point_eq x)⁻¹ ⬝ ap f p ⬝ point_eq y = idp :
      sigma_equiv_sigma_right (λp,
      calc point_eq x =[p] point_eq y ≃ point_eq x = ap f p ⬝ point_eq y : eq_pathover_equiv_Fl
           ... ≃ ap f p ⬝ point_eq y = point_eq x : eq_equiv_eq_symm
           ... ≃ (point_eq x)⁻¹ ⬝ (ap f p ⬝ point_eq y) = idp : eq_equiv_inv_con_eq_idp
           ... ≃ (point_eq x)⁻¹ ⬝ ap f p ⬝ point_eq y = idp : equiv_eq_closed_left _ !con.assoc⁻¹)
      ... ≃ fiber (ap1_gen f (point_eq x) (point_eq y)) (idpath b) : fiber.sigma_char

  definition loop_pfiber [constructor] {A B : Type*} (f : A →* B) : Ω (pfiber f) ≃* pfiber (Ω→ f) :=
  pequiv_of_equiv (fiber_eq_equiv_fiber pt pt)
    begin
      induction f with f f₀, induction B with B b₀, esimp at (f,f₀), induction f₀, reflexivity
    end

  definition point_fiber_eq_equiv_fiber {A B : Type} {f : A → B} {b : B} {x y : fiber f b}
    (p : x = y) : point (fiber_eq_equiv_fiber x y p) = ap1_gen point idp idp p :=
  by induction p; reflexivity

  lemma ppoint_loop_pfiber {A B : Type*} (f : A →* B) :
    ppoint (Ω→ f) ∘* loop_pfiber f ~* Ω→ (ppoint f) :=
  phomotopy.mk (point_fiber_eq_equiv_fiber)
    begin
     induction f with f f₀, induction B with B b₀, esimp at (f,f₀), induction f₀, reflexivity
    end

  lemma ppoint_loop_pfiber_inv {A B : Type*} (f : A →* B) :
    Ω→ (ppoint f) ∘* (loop_pfiber f)⁻¹ᵉ* ~* ppoint (Ω→ f) :=
  (phomotopy_pinv_right_of_phomotopy (ppoint_loop_pfiber f))⁻¹*

  -- rename to pfiber_pequiv_...
  lemma pfiber_equiv_of_phomotopy_ppoint {A B : Type*} {f g : A →* B} (h : f ~* g)
    : ppoint g ∘* pfiber_equiv_of_phomotopy h ~* ppoint f :=
  begin
    induction f with f f₀, induction g with g g₀, induction h with h h₀, induction B with B b₀,
    esimp at *, induction h₀, induction g₀,
    fapply phomotopy.mk,
    { reflexivity },
    { esimp [pfiber_equiv_of_phomotopy], exact !point_fiber_eq⁻¹ }
  end

  lemma pequiv_postcompose_ppoint {A B B' : Type*} (f : A →* B) (g : B ≃* B')
    : ppoint f ∘* fiber.pequiv_postcompose f g ~* ppoint (g ∘* f) :=
  begin
    induction f with f f₀, induction g with g hg g₀, induction B with B b₀,
    induction B' with B' b₀', esimp at *, induction g₀, induction f₀,
    fapply phomotopy.mk,
    { reflexivity },
    { esimp [pequiv_postcompose], symmetry,
      refine !ap_compose⁻¹ ⬝ _, apply ap_constant }
  end

  lemma pequiv_precompose_ppoint {A A' B : Type*} (f : A →* B) (g : A' ≃* A)
    : ppoint f ∘* fiber.pequiv_precompose f g ~* g ∘* ppoint (f ∘* g) :=
  begin
    induction f with f f₀, induction g with g hg g₀, induction B with B b₀,
    induction A with A a₀', esimp at *, induction g₀, induction f₀,
    reflexivity,
  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 :=
  begin
    refine !passoc⁻¹* ⬝* _,
    refine pwhisker_right _ !pequiv_precompose_ppoint ⬝* _,
    refine !passoc ⬝* _,
    apply pwhisker_left,
    refine !passoc⁻¹* ⬝* _,
    refine pwhisker_right _ !pfiber_equiv_of_phomotopy_ppoint ⬝* _,
    apply pinv_right_phomotopy_of_phomotopy,
    refine !pequiv_postcompose_ppoint⁻¹*,
  end

  definition is_trunc_fiber [instance] (n : ℕ₋₂) {A B : Type} (f : A → B) (b : B)
    [is_trunc n A] [is_trunc (n.+1) B] : is_trunc n (fiber f b) :=
  is_trunc_equiv_closed_rev n !fiber.sigma_char

  definition is_trunc_pfiber [instance] (n : ℕ₋₂) {A B : Type*} (f : A →* B)
    [is_trunc n A] [is_trunc (n.+1) B] : is_trunc n (pfiber f) :=
  is_trunc_fiber n f pt

  definition fiber_equiv_of_is_contr [constructor] {A B : Type} (f : A → B) (b : B) [is_contr B] :
    fiber f b ≃ A :=
  !fiber.sigma_char ⬝e !sigma_equiv_of_is_contr_right

  definition pfiber_pequiv_of_is_contr [constructor] {A B : Type*} (f : A →* B) [is_contr B] :
    pfiber f ≃* A :=
  pequiv_of_equiv (fiber_equiv_of_is_contr f pt) idp

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

  -- 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_succ_of_is_prop

end is_trunc

namespace is_conn

  open unit trunc_index nat is_trunc pointed.ops

  definition is_conn_equiv_closed_rev (n : ℕ₋₂) {A B : Type} (f : A ≃ B) (H : is_conn n B) :
    is_conn n A :=
  is_conn_equiv_closed n f⁻¹ᵉ _

  definition is_conn_succ_intro {n : ℕ₋₂} {A : Type} (a : trunc (n.+1) A)
    (H2 : Π(a a' : A), is_conn n (a = a')) : is_conn (n.+1) A :=
  begin
    apply @is_contr_of_inhabited_prop,
    { apply is_trunc_succ_intro,
      refine trunc.rec _, intro a, refine trunc.rec _, intro a',
      apply is_contr_equiv_closed !tr_eq_tr_equiv⁻¹ᵉ },
    exact a
  end

  definition is_conn_pathover (n : ℕ₋₂) {A : Type} {B : A → Type} {a a' : A} (p : a = a') (b : B a)
    (b' : B a') [is_conn (n.+1) (B a')] : is_conn n (b =[p] b') :=
  is_conn_equiv_closed_rev n !pathover_equiv_tr_eq _

  lemma is_conn_sigma [instance] {A : Type} (B : A → Type) (n : ℕ₋₂)
    [HA : is_conn n A] [HB : Πa, is_conn n (B a)] : is_conn n (Σa, B a) :=
  begin
    revert A B HA HB, induction n with n IH: intro A B HA HB,
    { apply is_conn_minus_two },
    apply is_conn_succ_intro,
    { induction center (trunc (n.+1) A) with a, induction center (trunc (n.+1) (B a)) with b,
      exact tr ⟨a, b⟩ },
    intro a a', refine is_conn_equiv_closed_rev n !sigma_eq_equiv _,
    apply IH, apply is_conn_eq, intro p, apply is_conn_pathover
    /- an alternative proof of the successor case -/
    -- induction center (trunc (n.+1) A) with a₀,
    -- induction center (trunc (n.+1) (B a₀)) with b₀,
    -- apply is_contr.mk (tr ⟨a₀, b₀⟩),
    -- intro ab, induction ab with ab, induction ab with a b,
    -- induction tr_eq_tr_equiv n a₀ a !is_prop.elim with p, induction p,
    -- induction tr_eq_tr_equiv n b₀ b !is_prop.elim with q, induction q,
    -- reflexivity
  end

  lemma is_conn_prod [instance] (A B : Type) (n : ℕ₋₂) [is_conn n A] [is_conn n B] :
    is_conn n (A × B) :=
  is_conn_equiv_closed n !sigma.equiv_prod _

  lemma is_conn_fun_of_is_conn {A B : Type} (n : ℕ₋₂) (f : A → B)
    [HA : is_conn n A] [HB : is_conn (n.+1) B] : is_conn_fun n f :=
  λb, is_conn_equiv_closed_rev n !fiber.sigma_char _

  lemma is_conn_pfiber {A B : Type*} (n : ℕ₋₂) (f : A →* B)
    [HA : is_conn n A] [HB : is_conn (n.+1) B] : is_conn n (pfiber f) :=
  is_conn_fun_of_is_conn n f pt

  definition is_conn_fun_trunc_elim_of_le {n k : ℕ₋₂} {A B : Type} [is_trunc n B] (f : A → B)
    (H : k ≤ n) [H2 : is_conn_fun k f] : is_conn_fun k (trunc.elim f : trunc n A → B) :=
  begin
    apply is_conn_fun.intro,
    intro P, have Πb, is_trunc n (P b), from (λb, is_trunc_of_le _ H),
    fconstructor,
    { intro f' b,
      refine is_conn_fun.elim k H2 _ _ b, intro a, exact f' (tr a) },
    { intro f', apply eq_of_homotopy, intro a,
      induction a with a, esimp, rewrite [is_conn_fun.elim_β] }
  end

  definition is_conn_fun_trunc_elim_of_ge {n k : ℕ₋₂} {A B : Type} [is_trunc n B] (f : A → B)
    (H : n ≤ k) [H2 : is_conn_fun k f] : is_conn_fun k (trunc.elim f : trunc n A → B) :=
  begin
   apply is_conn_fun_of_is_equiv,
   have H3 : is_equiv (trunc_functor k f), from !is_equiv_trunc_functor_of_is_conn_fun,
   have H4 : is_equiv (trunc_functor n f), from is_equiv_trunc_functor_of_le _ H,
   apply is_equiv_of_equiv_of_homotopy (equiv.mk (trunc_functor n f) _ ⬝e !trunc_equiv),
   intro x, induction x, reflexivity
  end

  definition is_conn_fun_trunc_elim {n k : ℕ₋₂} {A B : Type} [is_trunc n B] (f : A → B)
    [H2 : is_conn_fun k f] : is_conn_fun k (trunc.elim f : trunc n A → B) :=
  begin
    eapply algebra.le_by_cases k n: intro H,
    { exact is_conn_fun_trunc_elim_of_le f H },
    { exact is_conn_fun_trunc_elim_of_ge f H }
  end

  lemma is_conn_fun_tr (n : ℕ₋₂) (A : Type) : is_conn_fun n (tr : A → trunc n A) :=
  begin
    apply is_conn_fun.intro,
    intro P,
    fconstructor,
    { intro f' b, induction b with a, exact f' a },
    { intro f', reflexivity }
  end


  definition is_contr_of_is_conn_of_is_trunc {n : ℕ₋₂} {A : Type} (H : is_trunc n A)
    (K : is_conn n A) : is_contr A :=
  is_contr_equiv_closed (trunc_equiv n A)

  definition is_conn_fun_compose {n : ℕ₋₂} {A B C : Type} (g : B → C) (f : A → B)
    (H : is_conn_fun n g) (K : is_conn_fun n f) : is_conn_fun n (g ∘ f) :=
  sorry

  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

  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

  /- move! -/
  open sigma.ops pointed
  definition merely_constant {A B : Type} (f : A → B) : Type :=
  Σb, Πa, merely (f a = b)

  definition merely_constant_pmap {A B : Type*} {f : A →* B} (H : merely_constant f) (a : A) :
    merely (f a = pt) :=
  tconcat (tconcat (H.2 a) (tinverse (H.2 pt))) (tr (respect_pt f))

  definition merely_constant_of_is_conn {A B : Type*} (f : A →* B) [is_conn 0 A] : merely_constant f :=
  ⟨pt, is_conn.elim -1 _ (tr (respect_pt f))⟩

  open sigma
  definition component [constructor] (A : Type*) : Type* :=
  pType.mk (Σ(a : A), merely (pt = a)) ⟨pt, tr idp⟩

  lemma is_conn_component [instance] (A : Type*) : is_conn 0 (component A) :=
  is_contr.mk (tr pt)
    begin
      intro x, induction x with x, induction x with a p, induction p with p, induction p, reflexivity
    end

  definition component_incl [constructor] (A : Type*) : component A →* A :=
  pmap.mk pr1 idp

  definition is_embedding_component_incl [instance] (A : Type*) : is_embedding (component_incl A) :=
  is_embedding_pr1 _

  definition component_intro [constructor] {A B : Type*} (f : A →* B) (H : merely_constant f) :
    A →* component B :=
  begin
    fapply pmap.mk,
    { intro a, refine ⟨f a, _⟩, exact tinverse (merely_constant_pmap H a) },
    exact subtype_eq !respect_pt
  end

  definition component_functor [constructor] {A B : Type*} (f : A →* B) : component A →* component B :=
  component_intro (f ∘* component_incl A) !merely_constant_of_is_conn

  -- definition component_elim [constructor] {A B : Type*} (f : A →* B) (H : merely_constant f) :
  --   A →* component B :=
  -- begin
  --   fapply pmap.mk,
  --   { intro a, refine ⟨f a, _⟩, exact tinverse (merely_constant_pmap H a) },
  --   exact subtype_eq !respect_pt
  -- end

  definition loop_component (A : Type*) : Ω (component A) ≃* Ω A :=
  loop_pequiv_loop_of_is_embedding (component_incl A)

  lemma loopn_component (n : ℕ) (A : Type*) : Ω[n+1] (component A) ≃* Ω[n+1] A :=
  !loopn_succ_in ⬝e* loopn_pequiv_loopn n (loop_component A) ⬝e* !loopn_succ_in⁻¹ᵉ*

  -- lemma fundamental_group_component (A : Type*) : π₁ (component A) ≃g π₁ A :=
  -- isomorphism_of_equiv (trunc_equiv_trunc 0 (loop_component A)) _

  lemma homotopy_group_component (n : ℕ) (A : Type*) : πg[n+1] (component A) ≃g πg[n+1] A :=
  homotopy_group_isomorphism_of_is_embedding (n+1) (component_incl A)

  definition is_trunc_component [instance] (n : ℕ₋₂) (A : Type*) [is_trunc n A] :
    is_trunc n (component A) :=
  begin
    apply @is_trunc_sigma, intro a, cases n with n,
    { apply is_contr_of_inhabited_prop, exact tr !is_prop.elim },
    { apply is_trunc_succ_of_is_prop },
  end

  definition ptrunc_component' (n : ℕ₋₂) (A : Type*) :
    ptrunc (n.+2) (component A) ≃* component (ptrunc (n.+2) A) :=
  begin
    fapply pequiv.MK,
    { exact ptrunc.elim (n.+2) (component_functor !ptr) },
    { intro x, cases x with x p, induction x with a,
      refine tr ⟨a, _⟩,
      note q := trunc_functor -1 !tr_eq_tr_equiv p,
      exact trunc_trunc_equiv_left _ !minus_one_le_succ q },
    { exact sorry },
    { exact sorry }
  end

  definition ptrunc_component (n : ℕ₋₂) (A : Type*) :
    ptrunc n (component A) ≃* component (ptrunc n A) :=
  begin
    cases n with n, exact sorry,
    cases n with n, exact sorry,
    exact ptrunc_component' n A
  end

  definition pfiber_pequiv_component_of_is_contr [constructor] {A B : Type*} (f : A →* B) [is_contr B]
    /- extra condition, something like trunc_functor 0 f is an embedding -/ : pfiber f ≃* component A :=
  sorry

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 total_image {A B : Type} (f : A → B) : Type := sigma (image f)
local attribute is_prop.elim_set [recursor 6]
definition total_image.elim_set [unfold 8]
  {A B : Type} {f : A → B} {C : Type} [is_set C]
  (g : A → C) (h : Πa a', f a = f a' → g a = g a') (x : total_image f) : C :=
begin
  induction x with b v,
  induction v using is_prop.elim_set with x x x',
  { induction x with a p, exact g a },
  { induction x with a p, induction x' with a' p', induction p', exact h _ _ p }
end

definition total_image.rec [unfold 7]
  {A B : Type} {f : A → B} {C : total_image f → Type} [H : Πx, is_prop (C x)]
  (g : Πa, C ⟨f a, image.mk a idp⟩)
  (x : total_image f) : C x :=
begin
  induction x with b v,
  refine @image.rec _ _ _ _ _ (λv, H ⟨b, v⟩) _ v,
  intro a p,
  induction p, exact g a
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]

namespace int

  definition ring_int : Ring :=
  Ring.mk ℤ _

  notation `rℤ` := ring_int

  definition max0 : ℤ → ℕ
  | (of_nat n) := n
  | (-[1+ n])  := 0

  lemma le_max0 : Π(n : ℤ), n ≤ of_nat (max0 n)
  | (of_nat n) := proof le.refl n qed
  | (-[1+ n])  := proof unit.star qed

  lemma le_of_max0_le {n : ℤ} {m : ℕ} (h : max0 n ≤ m) : n ≤ of_nat m :=
  le.trans (le_max0 n) (of_nat_le_of_nat_of_le h)

end int

namespace set_quotient
  definition is_prop_set_quotient {A : Type} (R : A → A → Prop) [is_prop A] : is_prop (set_quotient R) :=
  begin
    apply is_prop.mk, intro x y,
    induction x using set_quotient.rec_prop, induction y using set_quotient.rec_prop,
    exact ap class_of !is_prop.elim
  end

  local attribute is_prop_set_quotient [instance]
  definition is_trunc_set_quotient [instance] (n : ℕ₋₂) {A : Type} (R : A → A → Prop) [is_trunc n A] :
    is_trunc n (set_quotient R) :=
  begin
    cases n with n, { apply is_contr_of_inhabited_prop, exact class_of !center },
    cases n with n, { apply _ },
    apply is_trunc_succ_succ_of_is_set
  end

  definition is_equiv_class_of [constructor] {A : Type} [is_set A] (R : A → A → Prop)
    (p : Π⦃a b⦄, R a b → a = b) : is_equiv (@class_of A R) :=
  begin
    fapply adjointify,
    { intro x, induction x, exact a, exact p H },
    { intro x, induction x using set_quotient.rec_prop, reflexivity },
    { intro a, reflexivity }
  end

  definition equiv_set_quotient [constructor] {A : Type} [is_set A] (R : A → A → Prop)
    (p : Π⦃a b⦄, R a b → a = b) : A ≃ set_quotient R :=
  equiv.mk _ (is_equiv_class_of R p)

end set_quotient

-- should be in pushout
namespace pushout
variables {TL BL TR : Type} (f : TL → BL) (g : TL → TR)

protected theorem elim_inl {P : Type} (Pinl : BL → P) (Pinr : TR → P)
  (Pglue : Π(x : TL), Pinl (f x) = Pinr (g x)) {b b' : BL} (p : b = b')
  : ap (pushout.elim Pinl Pinr Pglue) (ap inl p) = ap Pinl p :=
by cases p; reflexivity

protected theorem elim_inr {P : Type} (Pinl : BL → P) (Pinr : TR → P)
  (Pglue : Π(x : TL), Pinl (f x) = Pinr (g x)) {b b' : TR} (p : b = b')
  : ap (pushout.elim Pinl Pinr Pglue) (ap inr p) = ap Pinr p :=
by cases p; reflexivity

end pushout

-- should be in prod
namespace prod
open prod.ops
definition pair_eq_eta {A B : Type} {u v : A × B}
  (p : u = v) : pair_eq (p..1) (p..2) = prod.eta u ⬝ p ⬝ (prod.eta v)⁻¹ :=
by induction p; induction u; reflexivity

definition prod_eq_eq {A B : Type} {u v : A × B}
  {p₁ q₁ : u.1 = v.1} {p₂ q₂ : u.2 = v.2} (α₁ : p₁ = q₁) (α₂ : p₂ = q₂)
  : prod_eq p₁ p₂ = prod_eq q₁ q₂ :=
by cases α₁; cases α₂; reflexivity

definition prod_eq_assemble {A B : Type} {u v : A × B}
  {p q : u = v} (α₁ : p..1 = q..1) (α₂ : p..2 = q..2) : p = q :=
(prod_eq_eta p)⁻¹ ⬝ prod.prod_eq_eq α₁ α₂ ⬝ prod_eq_eta q

definition eq_pr1_concat {A B : Type} {u v w : A × B}
  (p : u = v) (q : v = w)
  : (p ⬝ q)..1 = p..1 ⬝ q..1 :=
by cases q; reflexivity

definition eq_pr2_concat {A B : Type} {u v w : A × B}
  (p : u = v) (q : v = w)
  : (p ⬝ q)..2 = p..2 ⬝ q..2 :=
by cases q; reflexivity

end prod