l3lab/ntp-mathlib huggingface dataset:
/--
Consider direct limits `lim G` and `lim G'` with direct system `f` and `f'` respectively, any
family of equivalences `eᵢ : Gᵢ ≅ G'ᵢ` such that `e ∘ f = f' ∘ e` induces an equivalence
`lim G ⟶ lim G'`.
-/
def congr [IsDirected ι (· ≤ ·)]
    (e : (i : ι) → G i ≃+ G' i)
    (he : ∀ i j h, (e j).toAddMonoidHom.comp (f i j h) = (f' i j h).comp (e i)) :
    DirectLimit G f ≃+ DirectLimit G' f' :=
  AddMonoidHom.toAddEquiv (map (e ·) he)
    (map (fun i ↦ (e i).symm) fun i j h ↦ FunLike.ext _ _ fun x ↦ by
      have eq1 := FunLike.congr_fun (he i j h) ((e i).symm x)
      simp only [AddMonoidHom.coe_comp, AddEquiv.coe_toAddMonoidHom, Function.comp_apply,
        AddMonoidHom.coe_coe, AddEquiv.apply_symm_apply] at eq1 ⊢
      simp [← eq1, of_f])
    (by 

Proof state:
R : Type u
inst✝⁶ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → AddCommGroup (G i)
f : (i j : ι) → i ≤ j → G i →+ G j
P : Type u₁
inst✝³ : AddCommGroup P
g : (i : ι) → G i →+ P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
G' : ι → Type v'
inst✝² : (i : ι) → AddCommGroup (G' i)
f' : (i j : ι) → i ≤ j → G' i →+ G' j
G'' : ι → Type v''
inst✝¹ : (i : ι) → AddCommGroup (G'' i)
f'' : (i j : ι) → i ≤ j → G'' i →+ G'' j
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
e : (i : ι) → G i ≃+ G' i
he :
  ∀ (i j : ι) (h : i ≤ j),
    AddMonoidHom.comp (AddEquiv.toAddMonoidHom (e j)) (f i j h) = AddMonoidHom.comp (f' i j h) ↑(e i)
⊢ AddMonoidHom.comp
      (map (fun i => ↑(AddEquiv.symm (e i)))
        (_ :
          ∀ (i j : ι) (h : i ≤ j),
            AddMonoidHom.comp ((fun i => ↑(AddEquiv.symm (e i))) j) (f' i j h) =
              AddMonoidHom.comp (f i j h) ((fun i => ↑(AddEquiv.symm (e i))) i)))
      (map (fun x => ↑(e x)) he) =
    AddMonoidHom.id (DirectLimit G f)

Next tactic:
rw [map_comp]
/--
Consider direct limits `lim G` and `lim G'` with direct system `f` and `f'` respectively, any
family of equivalences `eᵢ : Gᵢ ≅ G'ᵢ` such that `e ∘ f = f' ∘ e` induces an equivalence
`lim G ⟶ lim G'`.
-/
def congr [IsDirected ι (· ≤ ·)]
    (e : (i : ι) → G i ≃+ G' i)
    (he : ∀ i j h, (e j).toAddMonoidHom.comp (f i j h) = (f' i j h).comp (e i)) :
    DirectLimit G f ≃+ DirectLimit G' f' :=
  AddMonoidHom.toAddEquiv (map (e ·) he)
    (map (fun i ↦ (e i).symm) fun i j h ↦ FunLike.ext _ _ fun x ↦ by
      have eq1 := FunLike.congr_fun (he i j h) ((e i).symm x)
      simp only [AddMonoidHom.coe_comp, AddEquiv.coe_toAddMonoidHom, Function.comp_apply,
        AddMonoidHom.coe_coe, AddEquiv.apply_symm_apply] at eq1 ⊢
      simp [← eq1, of_f])
    (by rw [map_comp]; 

Proof state:
R : Type u
inst✝⁶ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → AddCommGroup (G i)
f : (i j : ι) → i ≤ j → G i →+ G j
P : Type u₁
inst✝³ : AddCommGroup P
g : (i : ι) → G i →+ P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
G' : ι → Type v'
inst✝² : (i : ι) → AddCommGroup (G' i)
f' : (i j : ι) → i ≤ j → G' i →+ G' j
G'' : ι → Type v''
inst✝¹ : (i : ι) → AddCommGroup (G'' i)
f'' : (i j : ι) → i ≤ j → G'' i →+ G'' j
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
e : (i : ι) → G i ≃+ G' i
he :
  ∀ (i j : ι) (h : i ≤ j),
    AddMonoidHom.comp (AddEquiv.toAddMonoidHom (e j)) (f i j h) = AddMonoidHom.comp (f' i j h) ↑(e i)
⊢ map (fun i => AddMonoidHom.comp ↑(AddEquiv.symm (e i)) ↑(e i))
      (_ :
        ∀ (i j : ι) (h : i ≤ j),
          AddMonoidHom.comp ((fun i => AddMonoidHom.comp ↑(AddEquiv.symm (e i)) ↑(e i)) j) (f i j h) =
            AddMonoidHom.comp (f i j h) ((fun i => AddMonoidHom.comp ↑(AddEquiv.symm (e i)) ↑(e i)) i)) =
    AddMonoidHom.id (DirectLimit G f)

Next tactic:
convert map_id
/--
Consider direct limits `lim G` and `lim G'` with direct system `f` and `f'` respectively, any
family of equivalences `eᵢ : Gᵢ ≅ G'ᵢ` such that `e ∘ f = f' ∘ e` induces an equivalence
`lim G ⟶ lim G'`.
-/
def congr [IsDirected ι (· ≤ ·)]
    (e : (i : ι) → G i ≃+ G' i)
    (he : ∀ i j h, (e j).toAddMonoidHom.comp (f i j h) = (f' i j h).comp (e i)) :
    DirectLimit G f ≃+ DirectLimit G' f' :=
  AddMonoidHom.toAddEquiv (map (e ·) he)
    (map (fun i ↦ (e i).symm) fun i j h ↦ FunLike.ext _ _ fun x ↦ by
      have eq1 := FunLike.congr_fun (he i j h) ((e i).symm x)
      simp only [AddMonoidHom.coe_comp, AddEquiv.coe_toAddMonoidHom, Function.comp_apply,
        AddMonoidHom.coe_coe, AddEquiv.apply_symm_apply] at eq1 ⊢
      simp [← eq1, of_f])
    (by rw [map_comp]; convert map_id <;> 

Proof state:
case h.e'_2.h.e'_10.h
R : Type u
inst✝⁶ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → AddCommGroup (G i)
f : (i j : ι) → i ≤ j → G i →+ G j
P : Type u₁
inst✝³ : AddCommGroup P
g : (i : ι) → G i →+ P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
G' : ι → Type v'
inst✝² : (i : ι) → AddCommGroup (G' i)
f' : (i j : ι) → i ≤ j → G' i →+ G' j
G'' : ι → Type v''
inst✝¹ : (i : ι) → AddCommGroup (G'' i)
f'' : (i j : ι) → i ≤ j → G'' i →+ G'' j
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
e : (i : ι) → G i ≃+ G' i
he :
  ∀ (i j : ι) (h : i ≤ j),
    AddMonoidHom.comp (AddEquiv.toAddMonoidHom (e j)) (f i j h) = AddMonoidHom.comp (f' i j h) ↑(e i)
x✝ : ι
⊢ AddMonoidHom.comp ↑(AddEquiv.symm (e x✝)) ↑(e x✝) = AddMonoidHom.id (G x✝)

Next tactic:
aesop
/--
Consider direct limits `lim G` and `lim G'` with direct system `f` and `f'` respectively, any
family of equivalences `eᵢ : Gᵢ ≅ G'ᵢ` such that `e ∘ f = f' ∘ e` induces an equivalence
`lim G ⟶ lim G'`.
-/
def congr [IsDirected ι (· ≤ ·)]
    (e : (i : ι) → G i ≃+ G' i)
    (he : ∀ i j h, (e j).toAddMonoidHom.comp (f i j h) = (f' i j h).comp (e i)) :
    DirectLimit G f ≃+ DirectLimit G' f' :=
  AddMonoidHom.toAddEquiv (map (e ·) he)
    (map (fun i ↦ (e i).symm) fun i j h ↦ FunLike.ext _ _ fun x ↦ by
      have eq1 := FunLike.congr_fun (he i j h) ((e i).symm x)
      simp only [AddMonoidHom.coe_comp, AddEquiv.coe_toAddMonoidHom, Function.comp_apply,
        AddMonoidHom.coe_coe, AddEquiv.apply_symm_apply] at eq1 ⊢
      simp [← eq1, of_f])
    (by rw [map_comp]; convert map_id <;> 

Proof state:
case convert_7
R : Type u
inst✝⁶ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → AddCommGroup (G i)
f : (i j : ι) → i ≤ j → G i →+ G j
P : Type u₁
inst✝³ : AddCommGroup P
g : (i : ι) → G i →+ P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
G' : ι → Type v'
inst✝² : (i : ι) → AddCommGroup (G' i)
f' : (i j : ι) → i ≤ j → G' i →+ G' j
G'' : ι → Type v''
inst✝¹ : (i : ι) → AddCommGroup (G'' i)
f'' : (i j : ι) → i ≤ j → G'' i →+ G'' j
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
e : (i : ι) → G i ≃+ G' i
he :
  ∀ (i j : ι) (h : i ≤ j),
    AddMonoidHom.comp (AddEquiv.toAddMonoidHom (e j)) (f i j h) = AddMonoidHom.comp (f' i j h) ↑(e i)
⊢ IsDirected ι fun x x_1 => x ≤ x_1

Next tactic:
aesop
/--
Consider direct limits `lim G` and `lim G'` with direct system `f` and `f'` respectively, any
family of equivalences `eᵢ : Gᵢ ≅ G'ᵢ` such that `e ∘ f = f' ∘ e` induces an equivalence
`lim G ⟶ lim G'`.
-/
def congr [IsDirected ι (· ≤ ·)]
    (e : (i : ι) → G i ≃+ G' i)
    (he : ∀ i j h, (e j).toAddMonoidHom.comp (f i j h) = (f' i j h).comp (e i)) :
    DirectLimit G f ≃+ DirectLimit G' f' :=
  AddMonoidHom.toAddEquiv (map (e ·) he)
    (map (fun i ↦ (e i).symm) fun i j h ↦ FunLike.ext _ _ fun x ↦ by
      have eq1 := FunLike.congr_fun (he i j h) ((e i).symm x)
      simp only [AddMonoidHom.coe_comp, AddEquiv.coe_toAddMonoidHom, Function.comp_apply,
        AddMonoidHom.coe_coe, AddEquiv.apply_symm_apply] at eq1 ⊢
      simp [← eq1, of_f])
    (by rw [map_comp]; convert map_id <;> aesop)
    (by 

Proof state:
R : Type u
inst✝⁶ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → AddCommGroup (G i)
f : (i j : ι) → i ≤ j → G i →+ G j
P : Type u₁
inst✝³ : AddCommGroup P
g : (i : ι) → G i →+ P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
G' : ι → Type v'
inst✝² : (i : ι) → AddCommGroup (G' i)
f' : (i j : ι) → i ≤ j → G' i →+ G' j
G'' : ι → Type v''
inst✝¹ : (i : ι) → AddCommGroup (G'' i)
f'' : (i j : ι) → i ≤ j → G'' i →+ G'' j
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
e : (i : ι) → G i ≃+ G' i
he :
  ∀ (i j : ι) (h : i ≤ j),
    AddMonoidHom.comp (AddEquiv.toAddMonoidHom (e j)) (f i j h) = AddMonoidHom.comp (f' i j h) ↑(e i)
⊢ AddMonoidHom.comp (map (fun x => ↑(e x)) he)
      (map (fun i => ↑(AddEquiv.symm (e i)))
        (_ :
          ∀ (i j : ι) (h : i ≤ j),
            AddMonoidHom.comp ((fun i => ↑(AddEquiv.symm (e i))) j) (f' i j h) =
              AddMonoidHom.comp (f i j h) ((fun i => ↑(AddEquiv.symm (e i))) i))) =
    AddMonoidHom.id (DirectLimit G' f')

Next tactic:
rw [map_comp]
/--
Consider direct limits `lim G` and `lim G'` with direct system `f` and `f'` respectively, any
family of equivalences `eᵢ : Gᵢ ≅ G'ᵢ` such that `e ∘ f = f' ∘ e` induces an equivalence
`lim G ⟶ lim G'`.
-/
def congr [IsDirected ι (· ≤ ·)]
    (e : (i : ι) → G i ≃+ G' i)
    (he : ∀ i j h, (e j).toAddMonoidHom.comp (f i j h) = (f' i j h).comp (e i)) :
    DirectLimit G f ≃+ DirectLimit G' f' :=
  AddMonoidHom.toAddEquiv (map (e ·) he)
    (map (fun i ↦ (e i).symm) fun i j h ↦ FunLike.ext _ _ fun x ↦ by
      have eq1 := FunLike.congr_fun (he i j h) ((e i).symm x)
      simp only [AddMonoidHom.coe_comp, AddEquiv.coe_toAddMonoidHom, Function.comp_apply,
        AddMonoidHom.coe_coe, AddEquiv.apply_symm_apply] at eq1 ⊢
      simp [← eq1, of_f])
    (by rw [map_comp]; convert map_id <;> aesop)
    (by rw [map_comp]; 

Proof state:
R : Type u
inst✝⁶ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → AddCommGroup (G i)
f : (i j : ι) → i ≤ j → G i →+ G j
P : Type u₁
inst✝³ : AddCommGroup P
g : (i : ι) → G i →+ P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
G' : ι → Type v'
inst✝² : (i : ι) → AddCommGroup (G' i)
f' : (i j : ι) → i ≤ j → G' i →+ G' j
G'' : ι → Type v''
inst✝¹ : (i : ι) → AddCommGroup (G'' i)
f'' : (i j : ι) → i ≤ j → G'' i →+ G'' j
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
e : (i : ι) → G i ≃+ G' i
he :
  ∀ (i j : ι) (h : i ≤ j),
    AddMonoidHom.comp (AddEquiv.toAddMonoidHom (e j)) (f i j h) = AddMonoidHom.comp (f' i j h) ↑(e i)
⊢ map (fun i => AddMonoidHom.comp ↑(e i) ↑(AddEquiv.symm (e i)))
      (_ :
        ∀ (i j : ι) (h : i ≤ j),
          AddMonoidHom.comp ((fun i => AddMonoidHom.comp ↑(e i) ↑(AddEquiv.symm (e i))) j) (f' i j h) =
            AddMonoidHom.comp (f' i j h) ((fun i => AddMonoidHom.comp ↑(e i) ↑(AddEquiv.symm (e i))) i)) =
    AddMonoidHom.id (DirectLimit G' f')

Next tactic:
convert map_id
/--
Consider direct limits `lim G` and `lim G'` with direct system `f` and `f'` respectively, any
family of equivalences `eᵢ : Gᵢ ≅ G'ᵢ` such that `e ∘ f = f' ∘ e` induces an equivalence
`lim G ⟶ lim G'`.
-/
def congr [IsDirected ι (· ≤ ·)]
    (e : (i : ι) → G i ≃+ G' i)
    (he : ∀ i j h, (e j).toAddMonoidHom.comp (f i j h) = (f' i j h).comp (e i)) :
    DirectLimit G f ≃+ DirectLimit G' f' :=
  AddMonoidHom.toAddEquiv (map (e ·) he)
    (map (fun i ↦ (e i).symm) fun i j h ↦ FunLike.ext _ _ fun x ↦ by
      have eq1 := FunLike.congr_fun (he i j h) ((e i).symm x)
      simp only [AddMonoidHom.coe_comp, AddEquiv.coe_toAddMonoidHom, Function.comp_apply,
        AddMonoidHom.coe_coe, AddEquiv.apply_symm_apply] at eq1 ⊢
      simp [← eq1, of_f])
    (by rw [map_comp]; convert map_id <;> aesop)
    (by rw [map_comp]; convert map_id <;> 

Proof state:
case h.e'_2.h.e'_10.h
R : Type u
inst✝⁶ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → AddCommGroup (G i)
f : (i j : ι) → i ≤ j → G i →+ G j
P : Type u₁
inst✝³ : AddCommGroup P
g : (i : ι) → G i →+ P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
G' : ι → Type v'
inst✝² : (i : ι) → AddCommGroup (G' i)
f' : (i j : ι) → i ≤ j → G' i →+ G' j
G'' : ι → Type v''
inst✝¹ : (i : ι) → AddCommGroup (G'' i)
f'' : (i j : ι) → i ≤ j → G'' i →+ G'' j
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
e : (i : ι) → G i ≃+ G' i
he :
  ∀ (i j : ι) (h : i ≤ j),
    AddMonoidHom.comp (AddEquiv.toAddMonoidHom (e j)) (f i j h) = AddMonoidHom.comp (f' i j h) ↑(e i)
x✝ : ι
⊢ AddMonoidHom.comp ↑(e x✝) ↑(AddEquiv.symm (e x✝)) = AddMonoidHom.id (G' x✝)

Next tactic:
aesop
/--
Consider direct limits `lim G` and `lim G'` with direct system `f` and `f'` respectively, any
family of equivalences `eᵢ : Gᵢ ≅ G'ᵢ` such that `e ∘ f = f' ∘ e` induces an equivalence
`lim G ⟶ lim G'`.
-/
def congr [IsDirected ι (· ≤ ·)]
    (e : (i : ι) → G i ≃+ G' i)
    (he : ∀ i j h, (e j).toAddMonoidHom.comp (f i j h) = (f' i j h).comp (e i)) :
    DirectLimit G f ≃+ DirectLimit G' f' :=
  AddMonoidHom.toAddEquiv (map (e ·) he)
    (map (fun i ↦ (e i).symm) fun i j h ↦ FunLike.ext _ _ fun x ↦ by
      have eq1 := FunLike.congr_fun (he i j h) ((e i).symm x)
      simp only [AddMonoidHom.coe_comp, AddEquiv.coe_toAddMonoidHom, Function.comp_apply,
        AddMonoidHom.coe_coe, AddEquiv.apply_symm_apply] at eq1 ⊢
      simp [← eq1, of_f])
    (by rw [map_comp]; convert map_id <;> aesop)
    (by rw [map_comp]; convert map_id <;> 

Proof state:
case convert_7
R : Type u
inst✝⁶ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → AddCommGroup (G i)
f : (i j : ι) → i ≤ j → G i →+ G j
P : Type u₁
inst✝³ : AddCommGroup P
g : (i : ι) → G i →+ P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
G' : ι → Type v'
inst✝² : (i : ι) → AddCommGroup (G' i)
f' : (i j : ι) → i ≤ j → G' i →+ G' j
G'' : ι → Type v''
inst✝¹ : (i : ι) → AddCommGroup (G'' i)
f'' : (i j : ι) → i ≤ j → G'' i →+ G'' j
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
e : (i : ι) → G i ≃+ G' i
he :
  ∀ (i j : ι) (h : i ≤ j),
    AddMonoidHom.comp (AddEquiv.toAddMonoidHom (e j)) (f i j h) = AddMonoidHom.comp (f' i j h) ↑(e i)
⊢ IsDirected ι fun x x_1 => x ≤ x_1

Next tactic:
aesop
lemma congr_symm_apply_of [IsDirected ι (· ≤ ·)]
    (e : (i : ι) → G i ≃+ G' i)
    (he : ∀ i j h, (e j).toAddMonoidHom.comp (f i j h) = (f' i j h).comp (e i))
    {i : ι} (g : G' i) :
    (congr e he).symm (of G' f' i g) = of G f i ((e i).symm g) := by
  

Proof state:
R : Type u
inst✝⁶ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → AddCommGroup (G i)
f : (i j : ι) → i ≤ j → G i →+ G j
P : Type u₁
inst✝³ : AddCommGroup P
g✝ : (i : ι) → G i →+ P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g✝ j) ((f i j hij) x) = (g✝ i) x
G' : ι → Type v'
inst✝² : (i : ι) → AddCommGroup (G' i)
f' : (i j : ι) → i ≤ j → G' i →+ G' j
G'' : ι → Type v''
inst✝¹ : (i : ι) → AddCommGroup (G'' i)
f'' : (i j : ι) → i ≤ j → G'' i →+ G'' j
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
e : (i : ι) → G i ≃+ G' i
he :
  ∀ (i j : ι) (h : i ≤ j),
    AddMonoidHom.comp (AddEquiv.toAddMonoidHom (e j)) (f i j h) = AddMonoidHom.comp (f' i j h) ↑(e i)
i : ι
g : G' i
⊢ (AddEquiv.symm (congr e he)) ((of G' f' i) g) = (of G f i) ((AddEquiv.symm (e i)) g)

Next tactic:
simp only [congr, AddMonoidHom.toAddEquiv_symm_apply, map_apply_of, AddMonoidHom.coe_coe]
instance zero : Zero (DirectLimit G f) := by
  

Proof state:
R : Type u
inst✝² : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝¹ : Preorder ι
G : ι → Type w
inst✝ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
⊢ Zero (DirectLimit G f)

Next tactic:
unfold DirectLimit
instance zero : Zero (DirectLimit G f) := by
  unfold DirectLimit
  

Proof state:
R : Type u
inst✝² : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝¹ : Preorder ι
G : ι → Type w
inst✝ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
⊢ Zero
    (FreeCommRing ((i : ι) × G i) ⧸
      Ideal.span
        {a |
          (∃ i j, ∃ (H : i ≤ j), ∃ x, of { fst := j, snd := f i j H x } - of { fst := i, snd := x } = a) ∨
            (∃ i, of { fst := i, snd := 1 } - 1 = a) ∨
              (∃ i x y, of { fst := i, snd := x + y } - (of { fst := i, snd := x } + of { fst := i, snd := y }) = a) ∨
                ∃ i x y, of { fst := i, snd := x * y } - of { fst := i, snd := x } * of { fst := i, snd := y } = a})

Next tactic:
exact ⟨0⟩
/-- Every element of the direct limit corresponds to some element in
some component of the directed system. -/
theorem exists_of [Nonempty ι] [IsDirected ι (· ≤ ·)] (z : DirectLimit G f) :
    ∃ i x, of G f i x = z :=
  Nonempty.elim (by 

Proof state:
R : Type u
inst✝⁴ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝³ : Preorder ι
G : ι → Type w
inst✝² : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
z : DirectLimit G f
⊢ Nonempty ι

Next tactic:
infer_instance
/-- Every element of the direct limit corresponds to some element in
some component of the directed system. -/
theorem exists_of [Nonempty ι] [IsDirected ι (· ≤ ·)] (z : DirectLimit G f) :
    ∃ i x, of G f i x = z :=
  Nonempty.elim (by infer_instance) fun ind : ι =>
    Quotient.inductionOn' z fun x =>
      FreeAbelianGroup.induction_on x ⟨ind, 0, (of _ _ ind).map_zero⟩
        (fun s =>
          Multiset.induction_on s ⟨ind, 1, (of _ _ ind).map_one⟩ fun a s ih =>
            let ⟨i, x⟩ := a
            let ⟨j, y, hs⟩ := ih
            let ⟨k, hik, hjk⟩ := exists_ge_ge i j
            ⟨k, f i k hik x * f j k hjk y, by
              

Proof state:
R : Type u
inst✝⁴ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝³ : Preorder ι
G : ι → Type w
inst✝² : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
z : DirectLimit G f
ind : ι
x✝ : FreeCommRing ((i : ι) × G i)
s✝ : Multiplicative (Multiset ((i : ι) × G i))
a : (i : ι) × G i
s : Multiset ((i : ι) × G i)
ih : ∃ i x, (of G f i) x = Quotient.mk'' (FreeAbelianGroup.of s)
i : ι
x : G i
j : ι
y : G j
hs : (of G f j) y = Quotient.mk'' (FreeAbelianGroup.of s)
k : ι
hik : i ≤ k
hjk : j ≤ k
⊢ (of G f k) (f i k hik x * f j k hjk y) = Quotient.mk'' (FreeAbelianGroup.of ({ fst := i, snd := x } ::ₘ s))

Next tactic:
rw [(of G f k).map_mul, of_f, of_f, hs]
/-- Every element of the direct limit corresponds to some element in
some component of the directed system. -/
theorem exists_of [Nonempty ι] [IsDirected ι (· ≤ ·)] (z : DirectLimit G f) :
    ∃ i x, of G f i x = z :=
  Nonempty.elim (by infer_instance) fun ind : ι =>
    Quotient.inductionOn' z fun x =>
      FreeAbelianGroup.induction_on x ⟨ind, 0, (of _ _ ind).map_zero⟩
        (fun s =>
          Multiset.induction_on s ⟨ind, 1, (of _ _ ind).map_one⟩ fun a s ih =>
            let ⟨i, x⟩ := a
            let ⟨j, y, hs⟩ := ih
            let ⟨k, hik, hjk⟩ := exists_ge_ge i j
            ⟨k, f i k hik x * f j k hjk y, by
              rw [(of G f k).map_mul, of_f, of_f, hs]
              /- porting note: In Lean3, from here, this was `by refl`. I have added
              the lemma `FreeCommRing.of_cons` to fix this proof. -/
              

Proof state:
R : Type u
inst✝⁴ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝³ : Preorder ι
G : ι → Type w
inst✝² : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
z : DirectLimit G f
ind : ι
x✝ : FreeCommRing ((i : ι) × G i)
s✝ : Multiplicative (Multiset ((i : ι) × G i))
a : (i : ι) × G i
s : Multiset ((i : ι) × G i)
ih : ∃ i x, (of G f i) x = Quotient.mk'' (FreeAbelianGroup.of s)
i : ι
x : G i
j : ι
y : G j
hs : (of G f j) y = Quotient.mk'' (FreeAbelianGroup.of s)
k : ι
hik : i ≤ k
hjk : j ≤ k
⊢ (of G f i) x * Quotient.mk'' (FreeAbelianGroup.of s) =
    Quotient.mk'' (FreeAbelianGroup.of ({ fst := i, snd := x } ::ₘ s))

Next tactic:
apply congr_arg Quotient.mk''
/-- Every element of the direct limit corresponds to some element in
some component of the directed system. -/
theorem exists_of [Nonempty ι] [IsDirected ι (· ≤ ·)] (z : DirectLimit G f) :
    ∃ i x, of G f i x = z :=
  Nonempty.elim (by infer_instance) fun ind : ι =>
    Quotient.inductionOn' z fun x =>
      FreeAbelianGroup.induction_on x ⟨ind, 0, (of _ _ ind).map_zero⟩
        (fun s =>
          Multiset.induction_on s ⟨ind, 1, (of _ _ ind).map_one⟩ fun a s ih =>
            let ⟨i, x⟩ := a
            let ⟨j, y, hs⟩ := ih
            let ⟨k, hik, hjk⟩ := exists_ge_ge i j
            ⟨k, f i k hik x * f j k hjk y, by
              rw [(of G f k).map_mul, of_f, of_f, hs]
              /- porting note: In Lean3, from here, this was `by refl`. I have added
              the lemma `FreeCommRing.of_cons` to fix this proof. -/
              apply congr_arg Quotient.mk''
              

Proof state:
R : Type u
inst✝⁴ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝³ : Preorder ι
G : ι → Type w
inst✝² : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
z : DirectLimit G f
ind : ι
x✝ : FreeCommRing ((i : ι) × G i)
s✝ : Multiplicative (Multiset ((i : ι) × G i))
a : (i : ι) × G i
s : Multiset ((i : ι) × G i)
ih : ∃ i x, (of G f i) x = Quotient.mk'' (FreeAbelianGroup.of s)
i : ι
x : G i
j : ι
y : G j
hs : (of G f j) y = Quotient.mk'' (FreeAbelianGroup.of s)
k : ι
hik : i ≤ k
hjk : j ≤ k
⊢ (fun x x_1 => x * x_1) (FreeCommRing.of { fst := i, snd := x }) (FreeAbelianGroup.of s) =
    FreeAbelianGroup.of ({ fst := i, snd := x } ::ₘ s)

Next tactic:
symm
/-- Every element of the direct limit corresponds to some element in
some component of the directed system. -/
theorem exists_of [Nonempty ι] [IsDirected ι (· ≤ ·)] (z : DirectLimit G f) :
    ∃ i x, of G f i x = z :=
  Nonempty.elim (by infer_instance) fun ind : ι =>
    Quotient.inductionOn' z fun x =>
      FreeAbelianGroup.induction_on x ⟨ind, 0, (of _ _ ind).map_zero⟩
        (fun s =>
          Multiset.induction_on s ⟨ind, 1, (of _ _ ind).map_one⟩ fun a s ih =>
            let ⟨i, x⟩ := a
            let ⟨j, y, hs⟩ := ih
            let ⟨k, hik, hjk⟩ := exists_ge_ge i j
            ⟨k, f i k hik x * f j k hjk y, by
              rw [(of G f k).map_mul, of_f, of_f, hs]
              /- porting note: In Lean3, from here, this was `by refl`. I have added
              the lemma `FreeCommRing.of_cons` to fix this proof. -/
              apply congr_arg Quotient.mk''
              symm
              

Proof state:
R : Type u
inst✝⁴ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝³ : Preorder ι
G : ι → Type w
inst✝² : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
z : DirectLimit G f
ind : ι
x✝ : FreeCommRing ((i : ι) × G i)
s✝ : Multiplicative (Multiset ((i : ι) × G i))
a : (i : ι) × G i
s : Multiset ((i : ι) × G i)
ih : ∃ i x, (of G f i) x = Quotient.mk'' (FreeAbelianGroup.of s)
i : ι
x : G i
j : ι
y : G j
hs : (of G f j) y = Quotient.mk'' (FreeAbelianGroup.of s)
k : ι
hik : i ≤ k
hjk : j ≤ k
⊢ FreeAbelianGroup.of ({ fst := i, snd := x } ::ₘ s) =
    (fun x x_1 => x * x_1) (FreeCommRing.of { fst := i, snd := x }) (FreeAbelianGroup.of s)

Next tactic:
apply FreeCommRing.of_cons