Spectral/algebra/submodule.hlean

/- submodules and quotient modules -/

-- Authors: Floris van Doorn, Jeremy Avigad
import .left_module .quotient_group

open algebra eq group sigma sigma.ops is_trunc function trunc equiv is_equiv property

  definition group_homomorphism_of_add_group_homomorphism [constructor] {G₁ G₂ : AddGroup}
    (φ : G₁ →a G₂) : G₁ →g G₂ :=
  φ

-- move to subgroup
-- attribute normal_subgroup_rel._trans_of_to_subgroup_rel [unfold 2]
-- attribute normal_subgroup_rel.to_subgroup_rel [constructor]

  definition is_equiv_incl_of_subgroup {G : Group} (H : property G) [is_subgroup G H] (h : Πg, g ∈ H) :
    is_equiv (incl_of_subgroup H) :=
  have is_surjective (incl_of_subgroup H),
  begin intro g, exact image.mk ⟨g, h g⟩ idp end,
  have is_embedding (incl_of_subgroup H), from is_embedding_incl_of_subgroup H,
  function.is_equiv_of_is_surjective_of_is_embedding (incl_of_subgroup H)

definition subgroup_isomorphism [constructor] {G : Group} (H : property G) [is_subgroup G H] (h : Πg, g ∈ H) :
  subgroup H ≃g G :=
isomorphism.mk _ (is_equiv_incl_of_subgroup H h)

definition is_equiv_qg_map {G : Group} (H : property G) [is_normal_subgroup G H] (H₂ : Π⦃g⦄, g ∈ H → g = 1) :
  is_equiv (qg_map H) :=
set_quotient.is_equiv_class_of _ (λg h r, eq_of_mul_inv_eq_one (H₂ r))

definition quotient_group_isomorphism [constructor] {G : Group} (H : property G) [is_normal_subgroup G H]
  (h : Πg, g ∈ H → g = 1) : quotient_group H ≃g G :=
(isomorphism.mk _ (is_equiv_qg_map H h))⁻¹ᵍ

definition is_equiv_ab_qg_map {G : AbGroup} (H : property G) [is_subgroup G H] (h : Π⦃g⦄, g ∈ H → g = 1) :
  is_equiv (ab_qg_map H) :=
proof @is_equiv_qg_map G H (is_normal_subgroup_ab _) h qed

definition ab_quotient_group_isomorphism [constructor] {G : AbGroup} (H : property G) [is_subgroup G H]
  (h : Πg, H g → g = 1) : quotient_ab_group H ≃g G :=
(isomorphism.mk _ (is_equiv_ab_qg_map H h))⁻¹ᵍ

namespace left_module
/- submodules -/
variables {R : Ring} {M M₁ M₂ M₃ : LeftModule R} {m m₁ m₂ : M}

structure is_submodule [class] (M : LeftModule R) (S : property M) : Type :=
  (zero_mem : 0 ∈ S)
  (add_mem : Π⦃g h⦄, g ∈ S → h ∈ S → g + h ∈ S)
  (smul_mem : Π⦃g⦄ (r : R), g ∈ S → r • g ∈ S)

definition zero_mem {R : Ring} {M : LeftModule R} (S : property M) [is_submodule M S] := is_submodule.zero_mem S
definition add_mem {R : Ring} {M : LeftModule R} (S : property M) [is_submodule M S] := @is_submodule.add_mem R M S
definition smul_mem {R : Ring} {M : LeftModule R} (S : property M) [is_submodule M S] := @is_submodule.smul_mem R M S

theorem neg_mem (S : property M) [is_submodule M S] ⦃m⦄ (H : m ∈ S) : -m ∈ S :=
transport (λx, x ∈ S) (neg_one_smul m) (smul_mem S (- 1) H)

theorem is_normal_submodule (S : property M) [is_submodule M S] ⦃m₁ m₂⦄ (H : S m₁) : S (m₂ + m₁ + (-m₂)) :=
transport (λx, S x) (by rewrite [add.comm, neg_add_cancel_left]) H

-- open is_submodule

variables {S : property M} [is_submodule M S]

definition is_subgroup_of_is_submodule [instance] (S : property M) [is_submodule M S] :
  is_subgroup (AddGroup_of_AddAbGroup M) S :=
is_subgroup.mk (zero_mem S) (add_mem S) (neg_mem S)

definition is_subgroup_of_is_submodule' [instance] (S : property M) [is_submodule M S] : is_subgroup (Group_of_AbGroup (AddAbGroup_of_LeftModule M)) S :=
is_subgroup.mk (zero_mem S) (add_mem S) (neg_mem S)

definition submodule' (S : property M) [is_submodule M S] : AddAbGroup :=
ab_subgroup S -- (subgroup_rel_of_submodule_rel S)

definition submodule_smul [constructor] (S : property M) [is_submodule M S] (r : R) :
  submodule' S →a submodule' S :=
ab_subgroup_functor (smul_homomorphism M r) (λg, smul_mem S r)

definition submodule_smul_right_distrib (r s : R) (n : submodule' S) :
  submodule_smul S (r + s) n = submodule_smul S r n + submodule_smul S s n :=
begin
  refine subgroup_functor_homotopy _ _ _ n ⬝ !subgroup_functor_mul⁻¹,
  intro m, exact to_smul_right_distrib r s m
end

definition submodule_mul_smul' (r s : R) (n : submodule' S) :
  submodule_smul S (r * s) n = (submodule_smul S r ∘g submodule_smul S s) n :=
begin
  refine subgroup_functor_homotopy _ _ _ n ⬝ (subgroup_functor_compose _ _ _ _ n)⁻¹ᵖ,
  intro m, exact to_mul_smul r s m
end

definition submodule_mul_smul (r s : R) (n : submodule' S) :
  submodule_smul S (r * s) n = submodule_smul S r (submodule_smul S s n) :=
by rexact submodule_mul_smul' r s n

definition submodule_one_smul (n : submodule' S) : submodule_smul S (1 : R) n = n :=
begin
  refine subgroup_functor_homotopy _ _ _ n ⬝ !subgroup_functor_gid,
  intro m, exact to_one_smul m
end

definition submodule (S : property M) [is_submodule M S] : LeftModule R :=
LeftModule_of_AddAbGroup (submodule' S) (submodule_smul S)
  (λr, homomorphism.addstruct (submodule_smul S r))
  submodule_smul_right_distrib
  submodule_mul_smul
  submodule_one_smul

definition submodule_incl [constructor] (S : property M) [is_submodule M S] : submodule S →lm M :=
lm_homomorphism_of_group_homomorphism (incl_of_subgroup _)
  begin
    intro r m, induction m with m hm, reflexivity
  end

definition hom_lift [constructor] {K : property M₂} [is_submodule M₂ K] (φ : M₁ →lm M₂)
  (h : Π (m : M₁), φ m ∈ K) : M₁ →lm submodule K :=
lm_homomorphism_of_group_homomorphism (hom_lift (group_homomorphism_of_lm_homomorphism φ) _ h)
  begin
    intro r g, exact subtype_eq (to_respect_smul φ r g)
  end

definition submodule_functor [constructor] {S : property M₁} [is_submodule M₁ S]
  {K : property M₂} [is_submodule M₂ K]
  (φ : M₁ →lm M₂) (h : Π (m : M₁), m ∈ S → φ m ∈ K) : submodule S →lm submodule K :=
hom_lift (φ ∘lm submodule_incl S) (by intro m; exact h m.1 m.2)

definition hom_lift_compose {K : property M₃} [is_submodule M₃ K]
  (φ : M₂ →lm M₃) (h : Π (m : M₂), φ m ∈ K) (ψ : M₁ →lm M₂) :
  hom_lift φ h ∘lm ψ ~ hom_lift (φ ∘lm ψ) proof (λm, h (ψ m)) qed :=
by reflexivity

definition hom_lift_homotopy {K : property M₂} [is_submodule M₂ K] {φ : M₁ →lm M₂}
  {h : Π (m : M₁), φ m ∈ K} {φ' : M₁ →lm M₂}
  {h' : Π (m : M₁), φ' m ∈ K} (p : φ ~ φ') : hom_lift φ h ~ hom_lift φ' h' :=
λg, subtype_eq (p g)

definition incl_smul (S : property M) [is_submodule M S] (r : R) (m : M) (h : S m) :
  r • ⟨m, h⟩ = ⟨_, smul_mem S r h⟩ :> submodule S :=
by reflexivity

definition property_submodule (S₁ S₂ : property M) [is_submodule M S₁] [is_submodule M S₂] :
  property (submodule S₁) := {m | submodule_incl S₁ m ∈ S₂}

definition is_submodule_property_submodule [instance] (S₁ S₂ : property M) [is_submodule M S₁] [is_submodule M S₂] :
  is_submodule (submodule S₁) (property_submodule S₁ S₂) :=
is_submodule.mk
  (mem_property_of (zero_mem S₂))
  (λm n p q, mem_property_of (add_mem S₂ (of_mem_property_of p) (of_mem_property_of q)))
  begin
    intro m r p, induction m with m hm, apply mem_property_of,
    apply smul_mem S₂, exact (of_mem_property_of p)
  end

definition eq_zero_of_mem_property_submodule_trivial [constructor] {S₁ S₂ : property M} [is_submodule M S₁] [is_submodule M S₂]
  (h : Π⦃m⦄, m ∈ S₂ → m = 0) ⦃m : submodule S₁⦄ (Sm : m ∈ property_submodule S₁ S₂) : m = 0 :=
begin
  fapply subtype_eq,
  apply h (of_mem_property_of Sm)
end

definition is_prop_submodule (S : property M) [is_submodule M S] [H : is_prop M] : is_prop (submodule S) :=
begin apply @is_trunc_sigma, exact H end
local attribute is_prop_submodule [instance]
definition is_contr_submodule [instance] (S : property M) [is_submodule M S] [is_contr M] : is_contr (submodule S) :=
is_contr_of_inhabited_prop 0

definition submodule_isomorphism [constructor] (S : property M) [is_submodule M S] (h : Πg, g ∈ S) :
  submodule S ≃lm M :=
isomorphism.mk (submodule_incl S) (is_equiv_incl_of_subgroup S h)

/- quotient modules -/

definition quotient_module' (S : property M) [is_submodule M S] : AddAbGroup :=
quotient_ab_group S -- (subgroup_rel_of_submodule_rel S)

definition quotient_module_smul [constructor] (S : property M) [is_submodule M S] (r : R) :
  quotient_module' S →a quotient_module' S :=
quotient_ab_group_functor (smul_homomorphism M r) (λg, smul_mem S r)

definition quotient_module_smul_right_distrib (r s : R) (n : quotient_module' S) :
  quotient_module_smul S (r + s) n = quotient_module_smul S r n + quotient_module_smul S s n :=
begin
  refine quotient_ab_group_functor_homotopy _ _ _ n ⬝ !quotient_ab_group_functor_mul⁻¹,
  intro m, exact to_smul_right_distrib r s m
end

definition quotient_module_mul_smul' (r s : R) (n : quotient_module' S) :
  quotient_module_smul S (r * s) n = (quotient_module_smul S r ∘g quotient_module_smul S s) n :=
begin
  apply eq.symm,
  apply eq.trans (quotient_ab_group_functor_compose _ _ _ _ n),
  apply quotient_ab_group_functor_homotopy,
  intro m, exact eq.symm (to_mul_smul r s m)
end
-- previous proof:
--  refine quotient_ab_group_functor_homotopy _ _ _ n ⬝
--    (quotient_ab_group_functor_compose (quotient_module_smul S r) (quotient_module_smul S s) _ _ n)⁻¹ᵖ,
--  intro m, to_mul_smul r s m

definition quotient_module_mul_smul (r s : R) (n : quotient_module' S) :
  quotient_module_smul S (r * s) n = quotient_module_smul S r (quotient_module_smul S s n) :=
by rexact quotient_module_mul_smul' r s n

definition quotient_module_one_smul (n : quotient_module' S) : quotient_module_smul S (1 : R) n = n :=
begin
  refine quotient_ab_group_functor_homotopy _ _ _ n ⬝ !quotient_ab_group_functor_gid,
  intro m, exact to_one_smul m
end

variable (S)
definition quotient_module (S : property M) [is_submodule M S] : LeftModule R :=
LeftModule_of_AddAbGroup (quotient_module' S) (quotient_module_smul S)
  (λr, homomorphism.addstruct (quotient_module_smul S r))
  quotient_module_smul_right_distrib
  quotient_module_mul_smul
  quotient_module_one_smul

definition quotient_map [constructor] : M →lm quotient_module S :=
lm_homomorphism_of_group_homomorphism (ab_qg_map _) (λr g, idp)

definition quotient_map_eq_zero (m : M) (H : S m) : quotient_map S m = 0 :=
@qg_map_eq_one _ _ (is_normal_subgroup_ab _) _ H

definition rel_of_quotient_map_eq_zero (m : M) (H : quotient_map S m = 0) : S m :=
@rel_of_qg_map_eq_one _ _ (is_normal_subgroup_ab _) m H

variable {S}
definition quotient_elim [constructor] (φ : M →lm M₂) (H : Π⦃m⦄, m ∈ S → φ m = 0) :
  quotient_module S →lm M₂ :=
lm_homomorphism_of_group_homomorphism
  (quotient_ab_group_elim (group_homomorphism_of_lm_homomorphism φ) H)
  begin
    intro r, esimp,
    refine @set_quotient.rec_prop _ _ _ (λ x, !is_trunc_eq) _,
    intro m,
    exact to_respect_smul φ r m
  end

definition is_prop_quotient_module (S : property M) [is_submodule M S] [H : is_prop M] : is_prop (quotient_module S) :=
begin apply @set_quotient.is_trunc_set_quotient, exact H end
local attribute is_prop_quotient_module [instance]

definition is_contr_quotient_module [instance] (S : property M) [is_submodule M S] [is_contr M] :
  is_contr (quotient_module S) :=
is_contr_of_inhabited_prop 0

definition rel_of_is_contr_quotient_module (S : property M) [is_submodule M S]
  (H : is_contr (quotient_module S)) (m : M) : S m :=
rel_of_quotient_map_eq_zero S m (@eq_of_is_contr _ H _ _)

definition quotient_module_isomorphism [constructor] (S : property M) [is_submodule M S] (h : Π⦃m⦄, S m → m = 0) :
  quotient_module S ≃lm M :=
(isomorphism.mk (quotient_map S) (is_equiv_ab_qg_map S h))⁻¹ˡᵐ

/- specific submodules -/
definition has_scalar_image (φ : M₁ →lm M₂) ⦃m : M₂⦄ (r : R)
  (h : image φ m) : image φ (r • m) :=
begin
  induction h with m' p,
  apply image.mk (r • m'),
  refine to_respect_smul φ r m' ⬝ ap (λx, r • x) p,
end

definition is_submodule_image [instance] (φ : M₁ →lm M₂) : is_submodule M₂ (image φ) :=
is_submodule.mk
  (show 0 ∈ image (group_homomorphism_of_lm_homomorphism φ),
    begin apply is_subgroup.one_mem, apply is_subgroup_image end)
  (λ g₁ g₂ hg₁ hg₂,
     show g₁ + g₂ ∈ image (group_homomorphism_of_lm_homomorphism φ),
     begin
       apply @is_subgroup.mul_mem,
       apply is_subgroup_image, exact hg₁, exact hg₂
     end)
  (has_scalar_image φ)

/-
definition image_rel [constructor] (φ : M₁ →lm M₂) : submodule_rel M₂ :=
submodule_rel_of_subgroup_rel
  (image_subgroup (group_homomorphism_of_lm_homomorphism φ))
  (has_scalar_image φ)
-/

definition image_trivial (φ : M₁ →lm M₂) [H : is_contr M₁] ⦃m : M₂⦄ (h : m ∈ image φ) : m = 0 :=
begin
  refine image.rec _ h,
  intro x p,
  refine p⁻¹ ⬝ ap φ _ ⬝ to_respect_zero φ,
  apply @is_prop.elim, apply is_trunc_succ, exact H
end

definition image_module [constructor] (φ : M₁ →lm M₂) : LeftModule R := submodule (image φ)

-- unfortunately this is note definitionally equal:
-- definition foo (φ : M₁ →lm M₂) :
--   (image_module φ : AddAbGroup) = image (group_homomorphism_of_lm_homomorphism φ) :=
-- by reflexivity

definition image_lift [constructor] (φ : M₁ →lm M₂) : M₁ →lm image_module φ :=
hom_lift φ (λm, image.mk m idp)

definition is_surjective_image_lift (φ : M₁ →lm M₂) : is_surjective (image_lift φ) :=
begin
  refine total_image.rec _, intro m, exact image.mk m (subtype_eq idp)
end

variables {ψ : M₂ →lm M₃} {φ : M₁ →lm M₂} {θ : M₁ →lm M₃}

definition image_elim [constructor] (θ : M₁ →lm M₃) (h : Π⦃g⦄, φ g = 0 → θ g = 0) :
  image_module φ →lm M₃ :=
begin
  fapply homomorphism.mk,
  change Image (group_homomorphism_of_lm_homomorphism φ) → M₃,
  exact image_elim (group_homomorphism_of_lm_homomorphism θ) h,
  split,
  { exact homomorphism.struct (image_elim (group_homomorphism_of_lm_homomorphism θ) _) },
  { intro r, refine @total_image.rec _ _ _ _ (λx, !is_trunc_eq) _, intro g,
    apply to_respect_smul }
end

definition image_elim_compute (h : Π⦃g⦄, φ g = 0 → θ g = 0) :
  image_elim θ h ∘lm image_lift φ ~ θ :=
begin
  reflexivity
end

-- definition image_elim_hom_lift (ψ : M →lm M₂) (h : Π⦃g⦄, φ g = 0 → θ g = 0) :
--   image_elim θ h ∘lm hom_lift ψ _ ~ _ :=
-- begin
--   reflexivity
-- end

definition is_contr_image_module [instance] (φ : M₁ →lm M₂) [is_contr M₂] :
  is_contr (image_module φ) :=
!is_contr_submodule

definition is_contr_image_module_of_is_contr_dom (φ : M₁ →lm M₂) [is_contrM₁ : is_contr M₁] :
  is_contr (image_module φ) :=
is_contr.mk 0
  begin
    have Π(x : image_module φ), is_prop (0 = x), from _,
    apply @total_image.rec,
    exact this,
    intro m,
    have h : is_contr (LeftModule.carrier M₁), from is_contrM₁,
    induction (eq_of_is_contr 0 m), apply subtype_eq,
    exact (to_respect_zero φ)⁻¹
  end

definition image_module_isomorphism [constructor] (φ : M₁ →lm M₂)
  (H : is_surjective φ) : image_module φ ≃lm M₂ :=
submodule_isomorphism _ H

definition has_scalar_kernel (φ : M₁ →lm M₂) ⦃m : M₁⦄ (r : R)
  (p : φ m = 0) : φ (r • m) = 0 :=
begin
  refine to_respect_smul φ r m ⬝ ap (λx, r • x) p ⬝ smul_zero r,
end

definition lm_kernel [reducible] (φ : M₁ →lm M₂) : property M₁ := kernel (group_homomorphism_of_lm_homomorphism φ)

definition is_submodule_kernel [instance] (φ : M₁ →lm M₂) : is_submodule M₁ (lm_kernel φ) :=
is_submodule.mk
  (show 0 ∈ kernel (group_homomorphism_of_lm_homomorphism φ),
    begin apply is_subgroup.one_mem, apply is_subgroup_kernel end)
  (λ g₁ g₂ hg₁ hg₂,
     show g₁ + g₂ ∈ kernel (group_homomorphism_of_lm_homomorphism φ),
       begin apply @is_subgroup.mul_mem, apply is_subgroup_kernel, exact hg₁, exact hg₂ end)
  (has_scalar_kernel φ)

definition kernel_full (φ : M₁ →lm M₂) [is_contr M₂] (m : M₁) : m ∈ lm_kernel φ :=
!is_prop.elim

definition kernel_module [reducible] (φ : M₁ →lm M₂) : LeftModule R := submodule (lm_kernel φ)

definition is_contr_kernel_module [instance] (φ : M₁ →lm M₂) [is_contr M₁] :
  is_contr (kernel_module φ) :=
!is_contr_submodule

definition kernel_module_isomorphism [constructor] (φ : M₁ →lm M₂) [is_contr M₂] : kernel_module φ ≃lm M₁ :=
submodule_isomorphism _ (kernel_full φ)

definition homology_quotient_property (ψ : M₂ →lm M₃) (φ : M₁ →lm M₂) : property (kernel_module ψ) :=
property_submodule (lm_kernel ψ) (image (homomorphism_fn φ))

definition is_submodule_homology_property [instance] (ψ : M₂ →lm M₃) (φ : M₁ →lm M₂) :
  is_submodule (kernel_module ψ) (homology_quotient_property ψ φ) :=
(is_submodule_property_submodule _ (image φ))

definition homology (ψ : M₂ →lm M₃) (φ : M₁ →lm M₂) : LeftModule R :=
quotient_module (homology_quotient_property ψ φ)

definition homology.mk (φ : M₁ →lm M₂) (m : M₂) (h : ψ m = 0) : homology ψ φ :=
quotient_map (homology_quotient_property ψ φ) ⟨m, h⟩

definition homology_eq0 {m : M₂} {hm : ψ m = 0} (h : image φ m) :
  homology.mk φ m hm = 0 :=
ab_qg_map_eq_one _ h

definition homology_eq0' {m : M₂} {hm : ψ m = 0} (h : image φ m):
  homology.mk φ m hm = homology.mk φ 0 (to_respect_zero ψ) :=
ab_qg_map_eq_one _ h

definition homology_eq {m n : M₂} {hm : ψ m = 0} {hn : ψ n = 0} (h : image φ (m - n)) :
  homology.mk φ m hm = homology.mk φ n hn :=
eq_of_sub_eq_zero (homology_eq0 h)

definition homology_elim [constructor] (θ : M₂ →lm M) (H : Πm, θ (φ m) = 0) :
  homology ψ φ →lm M :=
quotient_elim (θ ∘lm submodule_incl _)
  begin
    intro m x,
    induction m with m h,
    esimp at *,
    induction x with v,
    exact ap θ p⁻¹ ⬝ H v -- m'
  end

definition is_contr_homology [instance] (ψ : M₂ →lm M₃) (φ : M₁ →lm M₂) [is_contr M₂] :
  is_contr (homology ψ φ) :=
begin apply @is_contr_quotient_module end

definition homology_isomorphism [constructor] (ψ : M₂ →lm M₃) (φ : M₁ →lm M₂)
  [is_contr M₁] [is_contr M₃] : homology ψ φ ≃lm M₂ :=
(quotient_module_isomorphism (homology_quotient_property ψ φ)
  (eq_zero_of_mem_property_submodule_trivial (image_trivial _))) ⬝lm (kernel_module_isomorphism ψ)

definition ker_in_im_of_is_contr_homology (ψ : M₂ →lm M₃) {φ : M₁ →lm M₂}
  (H₁ : is_contr (homology ψ φ)) {m : M₂} (p : ψ m = 0) : image φ m :=
rel_of_is_contr_quotient_module _ H₁ ⟨m, p⟩

definition is_embedding_of_is_contr_homology_of_constant {ψ : M₂ →lm M₃} (φ : M₁ →lm M₂)
  (H₁ : is_contr (homology ψ φ)) (H₂ : Πm, φ m = 0) : is_embedding ψ :=
begin
  apply to_is_embedding_homomorphism (group_homomorphism_of_lm_homomorphism ψ),
  intro m p, note H := rel_of_is_contr_quotient_module _ H₁ ⟨m, p⟩,
  induction H with n q,
  exact q⁻¹ ⬝ H₂ n
end

definition is_embedding_of_is_contr_homology_of_is_contr {ψ : M₂ →lm M₃} (φ : M₁ →lm M₂)
  (H₁ : is_contr (homology ψ φ)) (H₂ : is_contr M₁) : is_embedding ψ :=
is_embedding_of_is_contr_homology_of_constant φ H₁
  (λm, ap φ (@eq_of_is_contr _ H₂ _ _) ⬝ respect_zero φ)

definition is_surjective_of_is_contr_homology_of_constant (ψ : M₂ →lm M₃) {φ : M₁ →lm M₂}
  (H₁ : is_contr (homology ψ φ)) (H₂ : Πm, ψ m = 0) : is_surjective φ :=
λm, ker_in_im_of_is_contr_homology ψ H₁ (H₂ m)

definition is_surjective_of_is_contr_homology_of_is_contr (ψ : M₂ →lm M₃) {φ : M₁ →lm M₂}
  (H₁ : is_contr (homology ψ φ)) (H₂ : is_contr M₃) : is_surjective φ :=
is_surjective_of_is_contr_homology_of_constant ψ H₁ (λm, @eq_of_is_contr _ H₂ _ _)

-- remove:

-- definition homology.rec (P : homology ψ φ → Type)
--   [H : Πx, is_set (P x)] (h₀ : Π(m : M₂) (h : ψ m = 0), P (homology.mk m h))
--   (h₁ : Π(m : M₂) (h : ψ m = 0) (k : image φ m), h₀ m h =[homology_eq0' k] h₀ 0 (to_respect_zero ψ))
--   : Πx, P x :=
-- begin
--   refine @set_quotient.rec _ _ _ H _ _,
--   { intro v, induction v with m h, exact h₀ m h },
--   { intro v v', induction v with m hm, induction v' with n hn,
--     intro h,
--     note x := h₁ (m - n) _ h,
--     esimp,
--     exact change_path _ _,
-- }
-- end

  -- definition quotient.rec (P : quotient_group N → Type)
  --   [H : Πx, is_set (P x)] (h₀ : Π(g : G), P (qg_map N g))
  --   -- (h₀_mul : Π(g h : G), h₀ (g * h))
  --   (h₁ : Π(g : G) (h : N g), h₀ g =[qg_map_eq_one g h] h₀ 1)
  --   : Πx, P x :=
  -- begin
  --   refine @set_quotient.rec _ _ _ H _ _,
  --   { intro g, exact h₀ g },
  --   { intro g g' h,
  --     note x := h₁ (g * g'⁻¹) h,
  --     }
  -- --   { intro v, induction  },
  -- --   { intro v v', induction v with m hm, induction v' with n hn,
  -- --     intro h,
  -- --     note x := h₁ (m - n) _ h,
  -- --     esimp,
  -- --     exact change_path _ _,
  -- -- }
  -- end



end left_module