l3lab/ntp-mathlib huggingface dataset:
/-- For `n ≤ m`, `(n, m)` is in the reflexive-transitive closure of `~` if `i ~ succ i`
  for all `i` between `n` and `m`. -/
theorem reflTransGen_of_succ_of_le (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ico n m, r i (succ i))
    (hnm : n ≤ m) : ReflTransGen r n m := by
  revert h; refine' Succ.rec _ _ hnm
  · intro _
    exact ReflTransGen.refl
  · intro m hnm ih h
    have : ReflTransGen r n m := ih fun i hi => h i ⟨hi.1, hi.2.trans_le <| le_succ m⟩
    rcases (le_succ m).eq_or_lt with hm | hm
    · 

Proof state:
case refine'_2.inl
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m✝ : α
hnm✝ : n ≤ m✝
m : α
hnm : n ≤ m
ih : (∀ i ∈ Ico n m, r i (succ i)) → ReflTransGen r n m
h : ∀ i ∈ Ico n (succ m), r i (succ i)
this : ReflTransGen r n m
hm : m = succ m
⊢ ReflTransGen r n (succ m)

Next tactic:
rwa [← hm]
/-- For `n ≤ m`, `(n, m)` is in the reflexive-transitive closure of `~` if `i ~ succ i`
  for all `i` between `n` and `m`. -/
theorem reflTransGen_of_succ_of_le (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ico n m, r i (succ i))
    (hnm : n ≤ m) : ReflTransGen r n m := by
  revert h; refine' Succ.rec _ _ hnm
  · intro _
    exact ReflTransGen.refl
  · intro m hnm ih h
    have : ReflTransGen r n m := ih fun i hi => h i ⟨hi.1, hi.2.trans_le <| le_succ m⟩
    rcases (le_succ m).eq_or_lt with hm | hm
    · rwa [← hm]
    

Proof state:
case refine'_2.inr
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m✝ : α
hnm✝ : n ≤ m✝
m : α
hnm : n ≤ m
ih : (∀ i ∈ Ico n m, r i (succ i)) → ReflTransGen r n m
h : ∀ i ∈ Ico n (succ m), r i (succ i)
this : ReflTransGen r n m
hm : m < succ m
⊢ ReflTransGen r n (succ m)

Next tactic:
exact this.tail (h m ⟨hnm, hm⟩)
/-- For `m ≤ n`, `(n, m)` is in the reflexive-transitive closure of `~` if `succ i ~ i`
  for all `i` between `n` and `m`. -/
theorem reflTransGen_of_succ_of_ge (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ico m n, r (succ i) i)
    (hmn : m ≤ n) : ReflTransGen r n m := by
  

Proof state:
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m : α
h : ∀ i ∈ Ico m n, r (succ i) i
hmn : m ≤ n
⊢ ReflTransGen r n m

Next tactic:
rw [← reflTransGen_swap]
/-- For `m ≤ n`, `(n, m)` is in the reflexive-transitive closure of `~` if `succ i ~ i`
  for all `i` between `n` and `m`. -/
theorem reflTransGen_of_succ_of_ge (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ico m n, r (succ i) i)
    (hmn : m ≤ n) : ReflTransGen r n m := by
  rw [← reflTransGen_swap]
  

Proof state:
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m : α
h : ∀ i ∈ Ico m n, r (succ i) i
hmn : m ≤ n
⊢ ReflTransGen (swap r) m n

Next tactic:
exact reflTransGen_of_succ_of_le (swap r) h hmn
/-- `(n, m)` is in the transitive closure of a reflexive relation `~` if `i ~ succ i` and
  `succ i ~ i` for all `i` between `n` and `m`. -/
theorem transGen_of_succ_of_reflexive (r : α → α → Prop) {n m : α} (hr : Reflexive r)
    (h1 : ∀ i ∈ Ico n m, r i (succ i)) (h2 : ∀ i ∈ Ico m n, r (succ i) i) : TransGen r n m := by
  

Proof state:
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m : α
hr : Reflexive r
h1 : ∀ i ∈ Ico n m, r i (succ i)
h2 : ∀ i ∈ Ico m n, r (succ i) i
⊢ TransGen r n m

Next tactic:
rcases eq_or_ne m n with (rfl | hmn)
/-- `(n, m)` is in the transitive closure of a reflexive relation `~` if `i ~ succ i` and
  `succ i ~ i` for all `i` between `n` and `m`. -/
theorem transGen_of_succ_of_reflexive (r : α → α → Prop) {n m : α} (hr : Reflexive r)
    (h1 : ∀ i ∈ Ico n m, r i (succ i)) (h2 : ∀ i ∈ Ico m n, r (succ i) i) : TransGen r n m := by
  rcases eq_or_ne m n with (rfl | hmn); · 

Proof state:
case inl
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
m : α
hr : Reflexive r
h1 : ∀ i ∈ Ico m m, r i (succ i)
h2 : ∀ i ∈ Ico m m, r (succ i) i
⊢ TransGen r m m

Next tactic:
exact TransGen.single (hr m)
/-- `(n, m)` is in the transitive closure of a reflexive relation `~` if `i ~ succ i` and
  `succ i ~ i` for all `i` between `n` and `m`. -/
theorem transGen_of_succ_of_reflexive (r : α → α → Prop) {n m : α} (hr : Reflexive r)
    (h1 : ∀ i ∈ Ico n m, r i (succ i)) (h2 : ∀ i ∈ Ico m n, r (succ i) i) : TransGen r n m := by
  rcases eq_or_ne m n with (rfl | hmn); · exact TransGen.single (hr m)
  

Proof state:
case inr
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m : α
hr : Reflexive r
h1 : ∀ i ∈ Ico n m, r i (succ i)
h2 : ∀ i ∈ Ico m n, r (succ i) i
hmn : m ≠ n
⊢ TransGen r n m

Next tactic:
exact transGen_of_succ_of_ne r h1 h2 hmn.symm
theorem exact_iff : Exact f g ↔ f.range = g.ker := by
  

Proof state:
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
⊢ Exact f g ↔ AddMonoidHom.range f = AddMonoidHom.ker g

Next tactic:
rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]
theorem exact_iff : Exact f g ↔ f.range = g.ker := by
  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]
  

Proof state:
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
⊢ f ≫ g = 0 ∧ Fork.ι (kernelCone g) ≫ Cofork.π (cokernelCocone f) = 0 ↔ AddMonoidHom.range f = AddMonoidHom.ker g

Next tactic:
exact
    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm
        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),
      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,
          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  

Proof state:
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
⊢ PreservesFiniteLimits colim

Next tactic:
refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  

Proof state:
case refine_1
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : Exact η γ
⊢ AddMonoidHom.range (colim.map η) ≤ AddMonoidHom.ker (colim.map γ)
case refine_2
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : Exact η γ
⊢ AddMonoidHom.ker (colim.map γ) ≤ AddMonoidHom.range (colim.map η)

Next tactic:
all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals 

Proof state:
case refine_1
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : Exact η γ
⊢ AddMonoidHom.range (colim.map η) ≤ AddMonoidHom.ker (colim.map γ)

Next tactic:
replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals 

Proof state:
case refine_2
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : Exact η γ
⊢ AddMonoidHom.ker (colim.map γ) ≤ AddMonoidHom.range (colim.map η)

Next tactic:
replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · 

Proof state:
case refine_1
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
⊢ AddMonoidHom.range (colim.map η) ≤ AddMonoidHom.ker (colim.map γ)

Next tactic:
rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
    

Proof state:
case refine_1
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
⊢ colim.map η ≫ colim.map γ = 0

Next tactic:
exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
    exact colimit.hom_ext fun j => by 

Proof state:
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
j : J
⊢ colimit.ι F j ≫ colim.map η ≫ colim.map γ = colimit.ι F j ≫ 0

Next tactic:
simp [reassoc_of% (h j).w]
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]
  · 

Proof state:
case refine_2
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
⊢ AddMonoidHom.ker (colim.map γ) ≤ AddMonoidHom.range (colim.map η)

Next tactic:
intro x (hx : _ = _)
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]
  · intro x (hx : _ = _)
    

Proof state:
case refine_2
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
x : ↑(colim.obj G)
hx : (colim.map γ) x = 0
⊢ x ∈ AddMonoidHom.range (colim.map η)

Next tactic:
rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]
  · intro x (hx : _ = _)
    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩
    

Proof state:
case refine_2.intro.intro
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
j : J
y : (forget AddCommGroupCat).obj (G.obj j)
hx : (colim.map γ) ((colimit.ι G j) y) = 0
⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)

Next tactic:
erw [← comp_apply, colimit.ι_map, comp_apply,
      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]
  · intro x (hx : _ = _)
    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩
    erw [← comp_apply, colimit.ι_map, comp_apply,
      ← map_zero (by 

Proof state:
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
j : J
y : (forget AddCommGroupCat).obj (G.obj j)
hx : (colimit.ι H j) ((γ.app j) y) = 0
⊢ ↑(H.obj j) →+ ↑(colimit H)

Next tactic:
exact colimit.ι H j
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]
  · intro x (hx : _ = _)
    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩
    erw [← comp_apply, colimit.ι_map, comp_apply,
      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx
    

Proof state:
case refine_2.intro.intro
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
j : J
y : (forget AddCommGroupCat).obj (G.obj j)
hx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0
⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)

Next tactic:
rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]
  · intro x (hx : _ = _)
    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩
    erw [← comp_apply, colimit.ι_map, comp_apply,
      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx
    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩
    

Proof state:
case refine_2.intro.intro.intro.intro.intro
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
j : J
y : (forget AddCommGroupCat).obj (G.obj j)
hx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0
k : J
e₁ e₂ : j ⟶ k
hk : (H.map e₁) ((γ.app j) y) = (H.map e₂) 0
⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)

Next tactic:
rw [map_zero, ← comp_apply, ← NatTrans.naturality, comp_apply] at hk
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]
  · intro x (hx : _ = _)
    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩
    erw [← comp_apply, colimit.ι_map, comp_apply,
      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx
    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩
    rw [map_zero, ← comp_apply, ← NatTrans.naturality, comp_apply] at hk
    

Proof state:
case refine_2.intro.intro.intro.intro.intro
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
j : J
y : (forget AddCommGroupCat).obj (G.obj j)
hx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0
k : J
e₁ e₂ : j ⟶ k
hk : (γ.app k) ((G.map e₁) y) = 0
⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)

Next tactic:
rcases ((exact_iff _ _).mp <| h k).ge hk with ⟨t, ht⟩
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]
  · intro x (hx : _ = _)
    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩
    erw [← comp_apply, colimit.ι_map, comp_apply,
      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx
    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩
    rw [map_zero, ← comp_apply, ← NatTrans.naturality, comp_apply] at hk
    rcases ((exact_iff _ _).mp <| h k).ge hk with ⟨t, ht⟩
    

Proof state:
case refine_2.intro.intro.intro.intro.intro.intro
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
j : J
y : (forget AddCommGroupCat).obj (G.obj j)
hx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0
k : J
e₁ e₂ : j ⟶ k
hk : (γ.app k) ((G.map e₁) y) = 0
t : ↑(F.obj k)
ht : (η.app k) t = (G.map e₁) y
⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)

Next tactic:
use colimit.ι F k t
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]
  · intro x (hx : _ = _)
    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩
    erw [← comp_apply, colimit.ι_map, comp_apply,
      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx
    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩
    rw [map_zero, ← comp_apply, ← NatTrans.naturality, comp_apply] at hk
    rcases ((exact_iff _ _).mp <| h k).ge hk with ⟨t, ht⟩
    use colimit.ι F k t
    

Proof state:
case h
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
j : J
y : (forget AddCommGroupCat).obj (G.obj j)
hx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0
k : J
e₁ e₂ : j ⟶ k
hk : (γ.app k) ((G.map e₁) y) = 0
t : ↑(F.obj k)
ht : (η.app k) t = (G.map e₁) y
⊢ (colim.map η) ((colimit.ι F k) t) = (colimit.ι G j) y

Next tactic:
erw [← comp_apply, colimit.ι_map, comp_apply, ht]
/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/
instance {J : Type u} [SmallCategory J] [IsFiltered J] :
    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by
  refine Functor.preservesFiniteLimitsOfMapExact _
    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)
  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=
    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h
  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]
    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]
  · intro x (hx : _ = _)
    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩
    erw [← comp_apply, colimit.ι_map, comp_apply,
      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx
    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩
    rw [map_zero, ← comp_apply, ← NatTrans.naturality, comp_apply] at hk
    rcases ((exact_iff _ _).mp <| h k).ge hk with ⟨t, ht⟩
    use colimit.ι F k t
    erw [← comp_apply, colimit.ι_map, comp_apply, ht]
    

Proof state:
case h
X Y Z : AddCommGroupCat
f : X ⟶ Y
g : Y ⟶ Z
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F G H : J ⥤ AddCommGroupCat
η : F ⟶ G
γ : G ⟶ H
h : ∀ (j : J), Exact (η.app j) (γ.app j)
j : J
y : (forget AddCommGroupCat).obj (G.obj j)
hx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0
k : J
e₁ e₂ : j ⟶ k
hk : (γ.app k) ((G.map e₁) y) = 0
t : ↑(F.obj k)
ht : (η.app k) t = (G.map e₁) y
⊢ (colimit.ι G k) ((G.map e₁) y) = (colimit.ι G j) y

Next tactic:
exact colimit.w_apply G e₁ y