l3lab/ntp-mathlib huggingface dataset:
/-- The normalized Moore complex functor, on morphisms.
-/
@[simps!]
def map (f : X ⟶ Y) : obj X ⟶ obj Y :=
  ChainComplex.ofHom _ _ _ _ _ _
    (fun n => factorThru _ (arrow _ ≫ f.app (op (SimplexCategory.mk n))) (by
      cases n <;> dsimp
      · apply top_factors
      · refine' (finset_inf_factors _).mpr fun i _ => kernelSubobject_factors _ _ _
        erw [Category.assoc, ← f.naturality,
          ← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ i (by simp)),
          Category.assoc, kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]))
    fun n => by
    cases n <;> dsimp [objD, objX] <;> 

Proof state:
case succ
C : Type u_1
inst✝¹ : Category.{?u.62442, u_1} C
inst✝ : Abelian C
X Y : SimplicialObject C
f✝ f : X ⟶ Y
n✝ : ℕ
⊢ factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Y (Fin.succ k)))
        (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ X (Fin.succ k))) ≫
          f.app (op (SimplexCategory.mk (Nat.succ n✝ + 1))))
        (_ :
          Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Y (Fin.succ k)))
            (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ X (Fin.succ k))) ≫
              f.app (op (SimplexCategory.mk (Nat.succ (Nat.succ n✝)))))) ≫
      factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Y (Fin.succ k)))
        (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Y (Fin.succ k))) ≫
          SimplicialObject.δ Y 0)
        (_ :
          Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Y (Fin.succ k)))
            (arrow (objX Y (n✝ + 1 + 1)) ≫ SimplicialObject.δ Y 0)) =
    factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ X (Fin.succ k)))
        (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ X (Fin.succ k))) ≫
          SimplicialObject.δ X 0)
        (_ :
          Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ X (Fin.succ k)))
            (arrow (objX X (n✝ + 1 + 1)) ≫ SimplicialObject.δ X 0)) ≫
      factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Y (Fin.succ k)))
        (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ X (Fin.succ k))) ≫
          f.app (op (SimplexCategory.mk (Nat.succ n✝))))
        (_ :
          Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Y (Fin.succ k)))
            (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ X (Fin.succ k))) ≫
              f.app (op (SimplexCategory.mk (Nat.succ n✝)))))

Next tactic:
aesop_cat
/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.

The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.

The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next.
-/
@[simps]
def normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := obj
  map f := map f
  -- Porting note: Why `aesop_cat` can't do `dsimp` steps?
  map_id X := by 

Proof state:
C : Type u_1
inst✝¹ : Category.{?u.107877, u_1} C
inst✝ : Abelian C
X : SimplicialObject C
⊢ { obj := obj, map := fun {X Y} f => map f }.map (𝟙 X) = 𝟙 ({ obj := obj, map := fun {X Y} f => map f }.obj X)

Next tactic:
ext (_ | _)
/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.

The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.

The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next.
-/
@[simps]
def normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := obj
  map f := map f
  -- Porting note: Why `aesop_cat` can't do `dsimp` steps?
  map_id X := by ext (_ | _) <;> 

Proof state:
case h.zero
C : Type u_1
inst✝¹ : Category.{?u.107877, u_1} C
inst✝ : Abelian C
X : SimplicialObject C
⊢ HomologicalComplex.Hom.f ({ obj := obj, map := fun {X Y} f => map f }.map (𝟙 X)) Nat.zero =
    HomologicalComplex.Hom.f (𝟙 ({ obj := obj, map := fun {X Y} f => map f }.obj X)) Nat.zero

Next tactic:
dsimp
/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.

The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.

The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next.
-/
@[simps]
def normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := obj
  map f := map f
  -- Porting note: Why `aesop_cat` can't do `dsimp` steps?
  map_id X := by ext (_ | _) <;> 

Proof state:
case h.succ
C : Type u_1
inst✝¹ : Category.{?u.107877, u_1} C
inst✝ : Abelian C
X : SimplicialObject C
n✝ : ℕ
⊢ HomologicalComplex.Hom.f ({ obj := obj, map := fun {X Y} f => map f }.map (𝟙 X)) (Nat.succ n✝) =
    HomologicalComplex.Hom.f (𝟙 ({ obj := obj, map := fun {X Y} f => map f }.obj X)) (Nat.succ n✝)

Next tactic:
dsimp
/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.

The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.

The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next.
-/
@[simps]
def normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := obj
  map f := map f
  -- Porting note: Why `aesop_cat` can't do `dsimp` steps?
  map_id X := by ext (_ | _) <;> dsimp <;> 

Proof state:
case h.zero
C : Type u_1
inst✝¹ : Category.{?u.107877, u_1} C
inst✝ : Abelian C
X : SimplicialObject C
⊢ Subobject.factorThru ⊤ (Subobject.arrow ⊤ ≫ 𝟙 (X.obj (op (SimplexCategory.mk 0))))
      (_ : Subobject.Factors (objX X 0) (Subobject.arrow (objX X 0) ≫ (𝟙 X).app (op (SimplexCategory.mk 0)))) =
    𝟙 (Subobject.underlying.obj ⊤)

Next tactic:
aesop_cat
/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.

The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.

The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next.
-/
@[simps]
def normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := obj
  map f := map f
  -- Porting note: Why `aesop_cat` can't do `dsimp` steps?
  map_id X := by ext (_ | _) <;> dsimp <;> 

Proof state:
case h.succ
C : Type u_1
inst✝¹ : Category.{?u.107877, u_1} C
inst✝ : Abelian C
X : SimplicialObject C
n✝ : ℕ
⊢ Subobject.factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ X (Fin.succ k)))
      (Subobject.arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ X (Fin.succ k))) ≫
        𝟙 (X.obj (op (SimplexCategory.mk (Nat.succ n✝)))))
      (_ :
        Subobject.Factors (objX X (Nat.succ n✝))
          (Subobject.arrow (objX X (Nat.succ n✝)) ≫ (𝟙 X).app (op (SimplexCategory.mk (Nat.succ n✝))))) =
    𝟙 (Subobject.underlying.obj (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ X (Fin.succ k))))

Next tactic:
aesop_cat
/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.

The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.

The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next.
-/
@[simps]
def normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := obj
  map f := map f
  -- Porting note: Why `aesop_cat` can't do `dsimp` steps?
  map_id X := by ext (_ | _) <;> dsimp <;> aesop_cat
  map_comp f g := by 

Proof state:
C : Type u_1
inst✝¹ : Category.{?u.107877, u_1} C
inst✝ : Abelian C
X✝ Y✝ Z✝ : SimplicialObject C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ { obj := obj, map := fun {X Y} f => map f }.map (f ≫ g) =
    { obj := obj, map := fun {X Y} f => map f }.map f ≫ { obj := obj, map := fun {X Y} f => map f }.map g

Next tactic:
ext (_ | _)
/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.

The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.

The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next.
-/
@[simps]
def normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := obj
  map f := map f
  -- Porting note: Why `aesop_cat` can't do `dsimp` steps?
  map_id X := by ext (_ | _) <;> dsimp <;> aesop_cat
  map_comp f g := by ext (_ | _) <;> 

Proof state:
case h.zero
C : Type u_1
inst✝¹ : Category.{?u.107877, u_1} C
inst✝ : Abelian C
X✝ Y✝ Z✝ : SimplicialObject C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ HomologicalComplex.Hom.f ({ obj := obj, map := fun {X Y} f => map f }.map (f ≫ g)) Nat.zero =
    HomologicalComplex.Hom.f
      ({ obj := obj, map := fun {X Y} f => map f }.map f ≫ { obj := obj, map := fun {X Y} f => map f }.map g) Nat.zero

Next tactic:
apply Subobject.eq_of_comp_arrow_eq
/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.

The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.

The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next.
-/
@[simps]
def normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := obj
  map f := map f
  -- Porting note: Why `aesop_cat` can't do `dsimp` steps?
  map_id X := by ext (_ | _) <;> dsimp <;> aesop_cat
  map_comp f g := by ext (_ | _) <;> 

Proof state:
case h.succ
C : Type u_1
inst✝¹ : Category.{?u.107877, u_1} C
inst✝ : Abelian C
X✝ Y✝ Z✝ : SimplicialObject C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
n✝ : ℕ
⊢ HomologicalComplex.Hom.f ({ obj := obj, map := fun {X Y} f => map f }.map (f ≫ g)) (Nat.succ n✝) =
    HomologicalComplex.Hom.f
      ({ obj := obj, map := fun {X Y} f => map f }.map f ≫ { obj := obj, map := fun {X Y} f => map f }.map g)
      (Nat.succ n✝)

Next tactic:
apply Subobject.eq_of_comp_arrow_eq
/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.

The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.

The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next.
-/
@[simps]
def normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := obj
  map f := map f
  -- Porting note: Why `aesop_cat` can't do `dsimp` steps?
  map_id X := by ext (_ | _) <;> dsimp <;> aesop_cat
  map_comp f g := by ext (_ | _) <;> apply Subobject.eq_of_comp_arrow_eq <;> 

Proof state:
case h.zero.h
C : Type u_1
inst✝¹ : Category.{?u.107877, u_1} C
inst✝ : Abelian C
X✝ Y✝ Z✝ : SimplicialObject C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ HomologicalComplex.Hom.f ({ obj := obj, map := fun {X Y} f => map f }.map (f ≫ g)) Nat.zero ≫
      Subobject.arrow (objX Z✝ Nat.zero) =
    HomologicalComplex.Hom.f
        ({ obj := obj, map := fun {X Y} f => map f }.map f ≫ { obj := obj, map := fun {X Y} f => map f }.map g)
        Nat.zero ≫
      Subobject.arrow (objX Z✝ Nat.zero)

Next tactic:
dsimp
/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.

The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.

The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next.
-/
@[simps]
def normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := obj
  map f := map f
  -- Porting note: Why `aesop_cat` can't do `dsimp` steps?
  map_id X := by ext (_ | _) <;> dsimp <;> aesop_cat
  map_comp f g := by ext (_ | _) <;> apply Subobject.eq_of_comp_arrow_eq <;> 

Proof state:
case h.succ.h
C : Type u_1
inst✝¹ : Category.{?u.107877, u_1} C
inst✝ : Abelian C
X✝ Y✝ Z✝ : SimplicialObject C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
n✝ : ℕ
⊢ HomologicalComplex.Hom.f ({ obj := obj, map := fun {X Y} f => map f }.map (f ≫ g)) (Nat.succ n✝) ≫
      Subobject.arrow (objX Z✝ (Nat.succ n✝)) =
    HomologicalComplex.Hom.f
        ({ obj := obj, map := fun {X Y} f => map f }.map f ≫ { obj := obj, map := fun {X Y} f => map f }.map g)
        (Nat.succ n✝) ≫
      Subobject.arrow (objX Z✝ (Nat.succ n✝))

Next tactic:
dsimp
/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.

The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.

The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next.
-/
@[simps]
def normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := obj
  map f := map f
  -- Porting note: Why `aesop_cat` can't do `dsimp` steps?
  map_id X := by ext (_ | _) <;> dsimp <;> aesop_cat
  map_comp f g := by ext (_ | _) <;> apply Subobject.eq_of_comp_arrow_eq <;> dsimp <;> 

Proof state:
case h.zero.h
C : Type u_1
inst✝¹ : Category.{?u.107877, u_1} C
inst✝ : Abelian C
X✝ Y✝ Z✝ : SimplicialObject C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ Subobject.factorThru ⊤ (Subobject.arrow ⊤ ≫ f.app (op (SimplexCategory.mk 0)) ≫ g.app (op (SimplexCategory.mk 0)))
        (_ : Subobject.Factors (objX Z✝ 0) (Subobject.arrow (objX X✝ 0) ≫ (f ≫ g).app (op (SimplexCategory.mk 0)))) ≫
      Subobject.arrow ⊤ =
    (Subobject.factorThru ⊤ (Subobject.arrow ⊤ ≫ f.app (op (SimplexCategory.mk 0)))
          (_ : Subobject.Factors (objX Y✝ 0) (Subobject.arrow (objX X✝ 0) ≫ f.app (op (SimplexCategory.mk 0)))) ≫
        Subobject.factorThru ⊤ (Subobject.arrow ⊤ ≫ g.app (op (SimplexCategory.mk 0)))
          (_ : Subobject.Factors (objX Z✝ 0) (Subobject.arrow (objX Y✝ 0) ≫ g.app (op (SimplexCategory.mk 0))))) ≫
      Subobject.arrow ⊤

Next tactic:
aesop_cat
/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.

The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.

The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next.
-/
@[simps]
def normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := obj
  map f := map f
  -- Porting note: Why `aesop_cat` can't do `dsimp` steps?
  map_id X := by ext (_ | _) <;> dsimp <;> aesop_cat
  map_comp f g := by ext (_ | _) <;> apply Subobject.eq_of_comp_arrow_eq <;> dsimp <;> 

Proof state:
case h.succ.h
C : Type u_1
inst✝¹ : Category.{?u.107877, u_1} C
inst✝ : Abelian C
X✝ Y✝ Z✝ : SimplicialObject C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
n✝ : ℕ
⊢ Subobject.factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Z✝ (Fin.succ k)))
        (Subobject.arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ X✝ (Fin.succ k))) ≫
          f.app (op (SimplexCategory.mk (Nat.succ n✝))) ≫ g.app (op (SimplexCategory.mk (Nat.succ n✝))))
        (_ :
          Subobject.Factors (objX Z✝ (Nat.succ n✝))
            (Subobject.arrow (objX X✝ (Nat.succ n✝)) ≫ (f ≫ g).app (op (SimplexCategory.mk (Nat.succ n✝))))) ≫
      Subobject.arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Z✝ (Fin.succ k))) =
    (Subobject.factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Y✝ (Fin.succ k)))
          (Subobject.arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ X✝ (Fin.succ k))) ≫
            f.app (op (SimplexCategory.mk (Nat.succ n✝))))
          (_ :
            Subobject.Factors (objX Y✝ (Nat.succ n✝))
              (Subobject.arrow (objX X✝ (Nat.succ n✝)) ≫ f.app (op (SimplexCategory.mk (Nat.succ n✝))))) ≫
        Subobject.factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Z✝ (Fin.succ k)))
          (Subobject.arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Y✝ (Fin.succ k))) ≫
            g.app (op (SimplexCategory.mk (Nat.succ n✝))))
          (_ :
            Subobject.Factors (objX Z✝ (Nat.succ n✝))
              (Subobject.arrow (objX Y✝ (Nat.succ n✝)) ≫ g.app (op (SimplexCategory.mk (Nat.succ n✝)))))) ≫
      Subobject.arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.δ Z✝ (Fin.succ k)))

Next tactic:
aesop_cat
instance : Category TopCommRingCat.{u} where
  Hom R S := { f : R →+* S // Continuous f }
  id R := ⟨RingHom.id R, by 

Proof state:
R : TopCommRingCat
⊢ Continuous ⇑(RingHom.id R.α)

Next tactic:
rw [RingHom.id]
instance : Category TopCommRingCat.{u} where
  Hom R S := { f : R →+* S // Continuous f }
  id R := ⟨RingHom.id R, by rw [RingHom.id]; 

Proof state:
R : TopCommRingCat
⊢ Continuous
    ⇑{
        toMonoidHom :=
          { toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },
            map_mul' :=
              (_ :
                ∀ (x y : R.α),
                  OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =
                    OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) },
        map_zero' :=
          (_ :
            OneHom.toFun
                (↑{ toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },
                    map_mul' :=
                      (_ :
                        ∀ (x y : R.α),
                          OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =
                            OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) })
                0 =
              OneHom.toFun
                (↑{ toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },
                    map_mul' :=
                      (_ :
                        ∀ (x y : R.α),
                          OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =
                            OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) })
                0),
        map_add' :=
          (_ :
            ∀ (x y : R.α),
              OneHom.toFun
                  (↑{ toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },
                      map_mul' :=
                        (_ :
                          ∀ (x y : R.α),
                            OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =
                              OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) })
                  (x + y) =
                OneHom.toFun
                  (↑{ toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },
                      map_mul' :=
                        (_ :
                          ∀ (x y : R.α),
                            OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =
                              OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) })
                  (x + y)) }

Next tactic:
continuity
instance : Category TopCommRingCat.{u} where
  Hom R S := { f : R →+* S // Continuous f }
  id R := ⟨RingHom.id R, by rw [RingHom.id]; continuity⟩
  comp f g :=
    ⟨g.val.comp f.val, by
      -- TODO automate
      

Proof state:
X✝ Y✝ Z✝ : TopCommRingCat
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ Continuous ⇑(RingHom.comp ↑g ↑f)

Next tactic:
cases f
instance : Category TopCommRingCat.{u} where
  Hom R S := { f : R →+* S // Continuous f }
  id R := ⟨RingHom.id R, by rw [RingHom.id]; continuity⟩
  comp f g :=
    ⟨g.val.comp f.val, by
      -- TODO automate
      cases f
      

Proof state:
case mk
X✝ Y✝ Z✝ : TopCommRingCat
g : Y✝ ⟶ Z✝
val✝ : X✝.α →+* Y✝.α
property✝ : Continuous ⇑val✝
⊢ Continuous ⇑(RingHom.comp ↑g ↑{ val := val✝, property := property✝ })

Next tactic:
cases g
instance : Category TopCommRingCat.{u} where
  Hom R S := { f : R →+* S // Continuous f }
  id R := ⟨RingHom.id R, by rw [RingHom.id]; continuity⟩
  comp f g :=
    ⟨g.val.comp f.val, by
      -- TODO automate
      cases f
      cases g
      

Proof state:
case mk.mk
X✝ Y✝ Z✝ : TopCommRingCat
val✝¹ : X✝.α →+* Y✝.α
property✝¹ : Continuous ⇑val✝¹
val✝ : Y✝.α →+* Z✝.α
property✝ : Continuous ⇑val✝
⊢ Continuous ⇑(RingHom.comp ↑{ val := val✝, property := property✝ } ↑{ val := val✝¹, property := property✝¹ })

Next tactic:
dsimp
instance : Category TopCommRingCat.{u} where
  Hom R S := { f : R →+* S // Continuous f }
  id R := ⟨RingHom.id R, by rw [RingHom.id]; continuity⟩
  comp f g :=
    ⟨g.val.comp f.val, by
      -- TODO automate
      cases f
      cases g
      dsimp; 

Proof state:
case mk.mk
X✝ Y✝ Z✝ : TopCommRingCat
val✝¹ : X✝.α →+* Y✝.α
property✝¹ : Continuous ⇑val✝¹
val✝ : Y✝.α →+* Z✝.α
property✝ : Continuous ⇑val✝
⊢ Continuous (⇑val✝ ∘ ⇑val✝¹)

Next tactic:
apply Continuous.comp
instance : Category TopCommRingCat.{u} where
  Hom R S := { f : R →+* S // Continuous f }
  id R := ⟨RingHom.id R, by rw [RingHom.id]; continuity⟩
  comp f g :=
    ⟨g.val.comp f.val, by
      -- TODO automate
      cases f
      cases g
      dsimp; apply Continuous.comp <;> 

Proof state:
case mk.mk.hg
X✝ Y✝ Z✝ : TopCommRingCat
val✝¹ : X✝.α →+* Y✝.α
property✝¹ : Continuous ⇑val✝¹
val✝ : Y✝.α →+* Z✝.α
property✝ : Continuous ⇑val✝
⊢ Continuous ⇑val✝

Next tactic:
assumption
instance : Category TopCommRingCat.{u} where
  Hom R S := { f : R →+* S // Continuous f }
  id R := ⟨RingHom.id R, by rw [RingHom.id]; continuity⟩
  comp f g :=
    ⟨g.val.comp f.val, by
      -- TODO automate
      cases f
      cases g
      dsimp; apply Continuous.comp <;> 

Proof state:
case mk.mk.hf
X✝ Y✝ Z✝ : TopCommRingCat
val✝¹ : X✝.α →+* Y✝.α
property✝¹ : Continuous ⇑val✝¹
val✝ : Y✝.α →+* Z✝.α
property✝ : Continuous ⇑val✝
⊢ Continuous ⇑val✝¹

Next tactic:
assumption
/-- The forgetful functors to `Type` do not reflect isomorphisms,
but the forgetful functor from `TopCommRingCat` to `TopCat` does.
-/
instance : ReflectsIsomorphisms (forget₂ TopCommRingCat.{u} TopCat.{u}) where
  reflects {X Y} f _ := by
    -- We have an isomorphism in `TopCat`,
    

Proof state:
X Y : TopCommRingCat
f : X ⟶ Y
x✝ : IsIso ((forget₂ TopCommRingCat TopCat).map f)
⊢ IsIso f

Next tactic:
let i_Top := asIso ((forget₂ TopCommRingCat TopCat).map f)
/-- The forgetful functors to `Type` do not reflect isomorphisms,
but the forgetful functor from `TopCommRingCat` to `TopCat` does.
-/
instance : ReflectsIsomorphisms (forget₂ TopCommRingCat.{u} TopCat.{u}) where
  reflects {X Y} f _ := by
    -- We have an isomorphism in `TopCat`,
    let i_Top := asIso ((forget₂ TopCommRingCat TopCat).map f)
    -- and a `RingEquiv`.
    

Proof state:
X Y : TopCommRingCat
f : X ⟶ Y
x✝ : IsIso ((forget₂ TopCommRingCat TopCat).map f)
i_Top : (forget₂ TopCommRingCat TopCat).obj X ≅ (forget₂ TopCommRingCat TopCat).obj Y :=
  asIso ((forget₂ TopCommRingCat TopCat).map f)
⊢ IsIso f

Next tactic:
let e_Ring : X ≃+* Y := { f.1, ((forget TopCat).mapIso i_Top).toEquiv with }
/-- The forgetful functors to `Type` do not reflect isomorphisms,
but the forgetful functor from `TopCommRingCat` to `TopCat` does.
-/
instance : ReflectsIsomorphisms (forget₂ TopCommRingCat.{u} TopCat.{u}) where
  reflects {X Y} f _ := by
    -- We have an isomorphism in `TopCat`,
    let i_Top := asIso ((forget₂ TopCommRingCat TopCat).map f)
    -- and a `RingEquiv`.
    let e_Ring : X ≃+* Y := { f.1, ((forget TopCat).mapIso i_Top).toEquiv with }
    -- Putting these together we obtain the isomorphism we're after:
    

Proof state:
X Y : TopCommRingCat
f : X ⟶ Y
x✝ : IsIso ((forget₂ TopCommRingCat TopCat).map f)
i_Top : (forget₂ TopCommRingCat TopCat).obj X ≅ (forget₂ TopCommRingCat TopCat).obj Y :=
  asIso ((forget₂ TopCommRingCat TopCat).map f)
e_Ring : X.α ≃+* Y.α :=
  let src := ↑f;
  let src_1 := ((forget TopCat).mapIso i_Top).toEquiv;
  {
    toEquiv :=
      { toFun := src.toFun, invFun := src_1.invFun, left_inv := (_ : Function.LeftInverse src_1.invFun src_1.toFun),
        right_inv := (_ : Function.RightInverse src_1.invFun src_1.toFun) },
    map_mul' := (_ : ∀ (x y : X.α), OneHom.toFun (↑↑src) (x * y) = OneHom.toFun (↑↑src) x * OneHom.toFun (↑↑src) y),
    map_add' := (_ : ∀ (x y : X.α), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) }
⊢ IsIso f

Next tactic:
exact
      ⟨⟨⟨e_Ring.symm, i_Top.inv.2⟩,
          ⟨by
            ext x
            exact e_Ring.left_inv x, by
            ext x
            exact e_Ring.right_inv x⟩⟩⟩