l3lab/ntp-mathlib huggingface dataset:
/-- Verify that the `pushout_cocone` is indeed the colimit. -/
def pushoutCoconeIsColimit : Limits.IsColimit (pushoutCocone f g) :=
  Limits.PushoutCocone.isColimitAux' _ fun s => by
    letI := RingHom.toAlgebra f
    letI := RingHom.toAlgebra g
    letI := RingHom.toAlgebra (f ≫ s.inl)
    let f' : A →ₐ[R] s.pt :=
      { s.inl with
        commutes' := fun r => rfl }
    let g' : B →ₐ[R] s.pt :=
      { s.inr with
        commutes' := fun r => by
          change (g ≫ s.inr) r = (f ≫ s.inl) r
          congr 1
          exact
            (s.ι.naturality Limits.WalkingSpan.Hom.snd).trans
              (s.ι.naturality Limits.WalkingSpan.Hom.fst).symm }
    -- Porting note : Lean has forget why `A ⊗[R] B` makes sense
    letI : Algebra R A := f.toAlgebra
    letI : Algebra R B := g.toAlgebra
    letI : Algebra R (pushoutCocone f g).pt := show Algebra R (A ⊗[R] B) by 

Proof state:
R A B : CommRingCat
f : R ⟶ A
g : R ⟶ B
s : PushoutCocone f g
this✝³ : Algebra ↑R ↑A := RingHom.toAlgebra f
this✝² : Algebra ↑R ↑B := RingHom.toAlgebra g
this✝¹ : Algebra ↑R ↑s.pt := RingHom.toAlgebra (f ≫ PushoutCocone.inl s)
f' : ↑A →ₐ[↑R] ↑s.pt :=
  let src := PushoutCocone.inl s;
  {
    toRingHom :=
      { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
        map_add' :=
          (_ : ∀ (x y : ↑A), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },
    commutes' :=
      (_ :
        ∀ (r : ↑R),
          OneHom.toFun
              (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
                    map_add' :=
                      (_ :
                        ∀ (x y : ↑A), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })
              ((algebraMap ↑R ↑A) r) =
            OneHom.toFun
              (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
                    map_add' :=
                      (_ :
                        ∀ (x y : ↑A), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })
              ((algebraMap ↑R ↑A) r)) }
g' : ↑B →ₐ[↑R] ↑s.pt :=
  let src := PushoutCocone.inr s;
  {
    toRingHom :=
      { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
        map_add' :=
          (_ : ∀ (x y : ↑B), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },
    commutes' := (_ : ∀ (r : ↑R), (g ≫ PushoutCocone.inr s) r = (f ≫ PushoutCocone.inl s) r) }
this✝ : Algebra ↑R ↑A := RingHom.toAlgebra f
this : Algebra ↑R ↑B := RingHom.toAlgebra g
⊢ Algebra (↑R) (↑A ⊗[↑R] ↑B)

Next tactic:
infer_instance
/-- Verify that the `pushout_cocone` is indeed the colimit. -/
def pushoutCoconeIsColimit : Limits.IsColimit (pushoutCocone f g) :=
  Limits.PushoutCocone.isColimitAux' _ fun s => by
    letI := RingHom.toAlgebra f
    letI := RingHom.toAlgebra g
    letI := RingHom.toAlgebra (f ≫ s.inl)
    let f' : A →ₐ[R] s.pt :=
      { s.inl with
        commutes' := fun r => rfl }
    let g' : B →ₐ[R] s.pt :=
      { s.inr with
        commutes' := fun r => by
          change (g ≫ s.inr) r = (f ≫ s.inl) r
          congr 1
          exact
            (s.ι.naturality Limits.WalkingSpan.Hom.snd).trans
              (s.ι.naturality Limits.WalkingSpan.Hom.fst).symm }
    -- Porting note : Lean has forget why `A ⊗[R] B` makes sense
    letI : Algebra R A := f.toAlgebra
    letI : Algebra R B := g.toAlgebra
    letI : Algebra R (pushoutCocone f g).pt := show Algebra R (A ⊗[R] B) by infer_instance
    -- The factor map is a ⊗ b ↦ f(a) * g(b).
    

Proof state:
R A B : CommRingCat
f : R ⟶ A
g : R ⟶ B
s : PushoutCocone f g
this✝⁴ : Algebra ↑R ↑A := RingHom.toAlgebra f
this✝³ : Algebra ↑R ↑B := RingHom.toAlgebra g
this✝² : Algebra ↑R ↑s.pt := RingHom.toAlgebra (f ≫ PushoutCocone.inl s)
f' : ↑A →ₐ[↑R] ↑s.pt :=
  let src := PushoutCocone.inl s;
  {
    toRingHom :=
      { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
        map_add' :=
          (_ : ∀ (x y : ↑A), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },
    commutes' :=
      (_ :
        ∀ (r : ↑R),
          OneHom.toFun
              (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
                    map_add' :=
                      (_ :
                        ∀ (x y : ↑A), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })
              ((algebraMap ↑R ↑A) r) =
            OneHom.toFun
              (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
                    map_add' :=
                      (_ :
                        ∀ (x y : ↑A), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })
              ((algebraMap ↑R ↑A) r)) }
g' : ↑B →ₐ[↑R] ↑s.pt :=
  let src := PushoutCocone.inr s;
  {
    toRingHom :=
      { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
        map_add' :=
          (_ : ∀ (x y : ↑B), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },
    commutes' := (_ : ∀ (r : ↑R), (g ≫ PushoutCocone.inr s) r = (f ≫ PushoutCocone.inl s) r) }
this✝¹ : Algebra ↑R ↑A := RingHom.toAlgebra f
this✝ : Algebra ↑R ↑B := RingHom.toAlgebra g
this : Algebra ↑R ↑(pushoutCocone f g).pt :=
  let_fun this := inferInstance;
  this
⊢ { l //
    PushoutCocone.inl (pushoutCocone f g) ≫ l = PushoutCocone.inl s ∧
      PushoutCocone.inr (pushoutCocone f g) ≫ l = PushoutCocone.inr s ∧
        ∀ {m : (pushoutCocone f g).pt ⟶ s.pt},
          PushoutCocone.inl (pushoutCocone f g) ≫ m = PushoutCocone.inl s →
            PushoutCocone.inr (pushoutCocone f g) ≫ m = PushoutCocone.inr s → m = l }

Next tactic:
use AlgHom.toRingHom (Algebra.TensorProduct.productMap f' g')
/-- The trivial ring is the (strict) terminal object of `CommRingCat`. -/
def punitIsTerminal : IsTerminal (CommRingCat.of.{u} PUnit) := by
  

Proof state:
⊢ IsTerminal (of PUnit.{u + 1})

Next tactic:
refine IsTerminal.ofUnique (h := fun X => ⟨⟨⟨⟨1, rfl⟩, fun _ _ => rfl⟩, ?_, ?_⟩, ?_⟩)
/-- The trivial ring is the (strict) terminal object of `CommRingCat`. -/
def punitIsTerminal : IsTerminal (CommRingCat.of.{u} PUnit) := by
  refine IsTerminal.ofUnique (h := fun X => ⟨⟨⟨⟨1, rfl⟩, fun _ _ => rfl⟩, ?_, ?_⟩, ?_⟩)
  · 

Proof state:
case refine_1
X : CommRingCat
⊢ OneHom.toFun
      (↑{ toOneHom := { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) },
          map_mul' :=
            (_ :
              ∀ (x x_1 : ↑X),
                OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1) =
                  OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1)) })
      0 =
    0

Next tactic:
dsimp
/-- The trivial ring is the (strict) terminal object of `CommRingCat`. -/
def punitIsTerminal : IsTerminal (CommRingCat.of.{u} PUnit) := by
  refine IsTerminal.ofUnique (h := fun X => ⟨⟨⟨⟨1, rfl⟩, fun _ _ => rfl⟩, ?_, ?_⟩, ?_⟩)
  · dsimp
  · 

Proof state:
case refine_2
X : CommRingCat
⊢ ∀ (x y : ↑X),
    OneHom.toFun
        (↑{ toOneHom := { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) },
            map_mul' :=
              (_ :
                ∀ (x x_1 : ↑X),
                  OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1) =
                    OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1)) })
        (x + y) =
      OneHom.toFun
          (↑{ toOneHom := { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) },
              map_mul' :=
                (_ :
                  ∀ (x x_1 : ↑X),
                    OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1) =
                      OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1)) })
          x +
        OneHom.toFun
          (↑{ toOneHom := { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) },
              map_mul' :=
                (_ :
                  ∀ (x x_1 : ↑X),
                    OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1) =
                      OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1)) })
          y

Next tactic:
intros
/-- The trivial ring is the (strict) terminal object of `CommRingCat`. -/
def punitIsTerminal : IsTerminal (CommRingCat.of.{u} PUnit) := by
  refine IsTerminal.ofUnique (h := fun X => ⟨⟨⟨⟨1, rfl⟩, fun _ _ => rfl⟩, ?_, ?_⟩, ?_⟩)
  · dsimp
  · intros; 

Proof state:
case refine_2
X : CommRingCat
x✝ y✝ : ↑X
⊢ OneHom.toFun
      (↑{ toOneHom := { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) },
          map_mul' :=
            (_ :
              ∀ (x x_1 : ↑X),
                OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1) =
                  OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1)) })
      (x✝ + y✝) =
    OneHom.toFun
        (↑{ toOneHom := { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) },
            map_mul' :=
              (_ :
                ∀ (x x_1 : ↑X),
                  OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1) =
                    OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1)) })
        x✝ +
      OneHom.toFun
        (↑{ toOneHom := { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) },
            map_mul' :=
              (_ :
                ∀ (x x_1 : ↑X),
                  OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1) =
                    OneHom.toFun { toFun := 1, map_one' := (_ : OfNat.ofNat 1 1 = OfNat.ofNat 1 1) } (x * x_1)) })
        y✝

Next tactic:
dsimp
/-- The trivial ring is the (strict) terminal object of `CommRingCat`. -/
def punitIsTerminal : IsTerminal (CommRingCat.of.{u} PUnit) := by
  refine IsTerminal.ofUnique (h := fun X => ⟨⟨⟨⟨1, rfl⟩, fun _ _ => rfl⟩, ?_, ?_⟩, ?_⟩)
  · dsimp
  · intros; dsimp
  · 

Proof state:
case refine_3
X : CommRingCat
⊢ ∀ (a : X ⟶ of PUnit.{u + 1}), a = default

Next tactic:
intros f
/-- The trivial ring is the (strict) terminal object of `CommRingCat`. -/
def punitIsTerminal : IsTerminal (CommRingCat.of.{u} PUnit) := by
  refine IsTerminal.ofUnique (h := fun X => ⟨⟨⟨⟨1, rfl⟩, fun _ _ => rfl⟩, ?_, ?_⟩, ?_⟩)
  · dsimp
  · intros; dsimp
  · intros f; 

Proof state:
case refine_3
X : CommRingCat
f : X ⟶ of PUnit.{u + 1}
⊢ f = default

Next tactic:
ext
/-- The trivial ring is the (strict) terminal object of `CommRingCat`. -/
def punitIsTerminal : IsTerminal (CommRingCat.of.{u} PUnit) := by
  refine IsTerminal.ofUnique (h := fun X => ⟨⟨⟨⟨1, rfl⟩, fun _ _ => rfl⟩, ?_, ?_⟩, ?_⟩)
  · dsimp
  · intros; dsimp
  · intros f; ext; 

Proof state:
case refine_3.w
X : CommRingCat
f : X ⟶ of PUnit.{u + 1}
x✝ : (forget CommRingCat).obj X
⊢ f x✝ = default x✝

Next tactic:
rfl
instance commRingCat_hasStrictTerminalObjects : HasStrictTerminalObjects CommRingCat.{u} := by
  

Proof state:
⊢ HasStrictTerminalObjects CommRingCat

Next tactic:
apply hasStrictTerminalObjects_of_terminal_is_strict (CommRingCat.of PUnit)
instance commRingCat_hasStrictTerminalObjects : HasStrictTerminalObjects CommRingCat.{u} := by
  apply hasStrictTerminalObjects_of_terminal_is_strict (CommRingCat.of PUnit)
  

Proof state:
⊢ ∀ (A : CommRingCat) (f : of PUnit.{u + 1} ⟶ A), IsIso f

Next tactic:
intro X f
instance commRingCat_hasStrictTerminalObjects : HasStrictTerminalObjects CommRingCat.{u} := by
  apply hasStrictTerminalObjects_of_terminal_is_strict (CommRingCat.of PUnit)
  intro X f
  

Proof state:
X : CommRingCat
f : of PUnit.{u + 1} ⟶ X
⊢ IsIso f

Next tactic:
refine ⟨⟨⟨1, rfl, fun _ _ => rfl⟩, by ext; rfl, ?_⟩⟩
instance commRingCat_hasStrictTerminalObjects : HasStrictTerminalObjects CommRingCat.{u} := by
  apply hasStrictTerminalObjects_of_terminal_is_strict (CommRingCat.of PUnit)
  intro X f
  refine ⟨⟨⟨1, rfl, fun _ _ => rfl⟩, by 

Proof state:
X : CommRingCat
f : of PUnit.{u + 1} ⟶ X
⊢ f ≫
      { toMonoidHom := 1, map_zero' := (_ : OneHom.toFun (↑1) 0 = OneHom.toFun (↑1) 0),
        map_add' := (_ : ∀ (x x_1 : ↑X), OneHom.toFun (↑1) (x + x_1) = OneHom.toFun (↑1) (x + x_1)) } =
    𝟙 (of PUnit.{u + 1})

Next tactic:
ext
instance commRingCat_hasStrictTerminalObjects : HasStrictTerminalObjects CommRingCat.{u} := by
  apply hasStrictTerminalObjects_of_terminal_is_strict (CommRingCat.of PUnit)
  intro X f
  refine ⟨⟨⟨1, rfl, fun _ _ => rfl⟩, by ext; 

Proof state:
case w
X : CommRingCat
f : of PUnit.{u + 1} ⟶ X
x✝ : (forget CommRingCat).obj (of PUnit.{u + 1})
⊢ (f ≫
        { toMonoidHom := 1, map_zero' := (_ : OneHom.toFun (↑1) 0 = OneHom.toFun (↑1) 0),
          map_add' := (_ : ∀ (x x_1 : ↑X), OneHom.toFun (↑1) (x + x_1) = OneHom.toFun (↑1) (x + x_1)) })
      x✝ =
    (𝟙 (of PUnit.{u + 1})) x✝

Next tactic:
rfl
instance commRingCat_hasStrictTerminalObjects : HasStrictTerminalObjects CommRingCat.{u} := by
  apply hasStrictTerminalObjects_of_terminal_is_strict (CommRingCat.of PUnit)
  intro X f
  refine ⟨⟨⟨1, rfl, fun _ _ => rfl⟩, by ext; rfl, ?_⟩⟩
  

Proof state:
X : CommRingCat
f : of PUnit.{u + 1} ⟶ X
⊢ { toMonoidHom := 1, map_zero' := (_ : OneHom.toFun (↑1) 0 = OneHom.toFun (↑1) 0),
        map_add' := (_ : ∀ (x x_1 : ↑X), OneHom.toFun (↑1) (x + x_1) = OneHom.toFun (↑1) (x + x_1)) } ≫
      f =
    𝟙 X

Next tactic:
ext x
instance commRingCat_hasStrictTerminalObjects : HasStrictTerminalObjects CommRingCat.{u} := by
  apply hasStrictTerminalObjects_of_terminal_is_strict (CommRingCat.of PUnit)
  intro X f
  refine ⟨⟨⟨1, rfl, fun _ _ => rfl⟩, by ext; rfl, ?_⟩⟩
  ext x
  

Proof state:
case w
X : CommRingCat
f : of PUnit.{u + 1} ⟶ X
x : (forget CommRingCat).obj X
⊢ ({ toMonoidHom := 1, map_zero' := (_ : OneHom.toFun (↑1) 0 = OneHom.toFun (↑1) 0),
          map_add' := (_ : ∀ (x x_1 : ↑X), OneHom.toFun (↑1) (x + x_1) = OneHom.toFun (↑1) (x + x_1)) } ≫
        f)
      x =
    (𝟙 X) x

Next tactic:
have e : (0 : X) = 1 := by
    rw [← f.map_one, ← f.map_zero]
    congr
instance commRingCat_hasStrictTerminalObjects : HasStrictTerminalObjects CommRingCat.{u} := by
  apply hasStrictTerminalObjects_of_terminal_is_strict (CommRingCat.of PUnit)
  intro X f
  refine ⟨⟨⟨1, rfl, fun _ _ => rfl⟩, by ext; rfl, ?_⟩⟩
  ext x
  have e : (0 : X) = 1 := by
    

Proof state:
X : CommRingCat
f : of PUnit.{u + 1} ⟶ X
x : (forget CommRingCat).obj X
⊢ 0 = 1

Next tactic:
rw [← f.map_one, ← f.map_zero]
instance commRingCat_hasStrictTerminalObjects : HasStrictTerminalObjects CommRingCat.{u} := by
  apply hasStrictTerminalObjects_of_terminal_is_strict (CommRingCat.of PUnit)
  intro X f
  refine ⟨⟨⟨1, rfl, fun _ _ => rfl⟩, by ext; rfl, ?_⟩⟩
  ext x
  have e : (0 : X) = 1 := by
    rw [← f.map_one, ← f.map_zero]
    

Proof state:
X : CommRingCat
f : of PUnit.{u + 1} ⟶ X
x : (forget CommRingCat).obj X
⊢ f 0 = f 1

Next tactic:
congr
instance commRingCat_hasStrictTerminalObjects : HasStrictTerminalObjects CommRingCat.{u} := by
  apply hasStrictTerminalObjects_of_terminal_is_strict (CommRingCat.of PUnit)
  intro X f
  refine ⟨⟨⟨1, rfl, fun _ _ => rfl⟩, by ext; rfl, ?_⟩⟩
  ext x
  have e : (0 : X) = 1 := by
    rw [← f.map_one, ← f.map_zero]
    congr
  

Proof state:
case w
X : CommRingCat
f : of PUnit.{u + 1} ⟶ X
x : (forget CommRingCat).obj X
e : 0 = 1
⊢ ({ toMonoidHom := 1, map_zero' := (_ : OneHom.toFun (↑1) 0 = OneHom.toFun (↑1) 0),
          map_add' := (_ : ∀ (x x_1 : ↑X), OneHom.toFun (↑1) (x + x_1) = OneHom.toFun (↑1) (x + x_1)) } ≫
        f)
      x =
    (𝟙 X) x

Next tactic:
replace e : 0 * x = 1 * x := congr_arg (· * x) e
instance commRingCat_hasStrictTerminalObjects : HasStrictTerminalObjects CommRingCat.{u} := by
  apply hasStrictTerminalObjects_of_terminal_is_strict (CommRingCat.of PUnit)
  intro X f
  refine ⟨⟨⟨1, rfl, fun _ _ => rfl⟩, by ext; rfl, ?_⟩⟩
  ext x
  have e : (0 : X) = 1 := by
    rw [← f.map_one, ← f.map_zero]
    congr
  replace e : 0 * x = 1 * x := congr_arg (· * x) e
  

Proof state:
case w
X : CommRingCat
f : of PUnit.{u + 1} ⟶ X
x : (forget CommRingCat).obj X
e : 0 * x = 1 * x
⊢ ({ toMonoidHom := 1, map_zero' := (_ : OneHom.toFun (↑1) 0 = OneHom.toFun (↑1) 0),
          map_add' := (_ : ∀ (x x_1 : ↑X), OneHom.toFun (↑1) (x + x_1) = OneHom.toFun (↑1) (x + x_1)) } ≫
        f)
      x =
    (𝟙 X) x

Next tactic:
rw [one_mul, zero_mul, ← f.map_zero] at e
instance commRingCat_hasStrictTerminalObjects : HasStrictTerminalObjects CommRingCat.{u} := by
  apply hasStrictTerminalObjects_of_terminal_is_strict (CommRingCat.of PUnit)
  intro X f
  refine ⟨⟨⟨1, rfl, fun _ _ => rfl⟩, by ext; rfl, ?_⟩⟩
  ext x
  have e : (0 : X) = 1 := by
    rw [← f.map_one, ← f.map_zero]
    congr
  replace e : 0 * x = 1 * x := congr_arg (· * x) e
  rw [one_mul, zero_mul, ← f.map_zero] at e
  

Proof state:
case w
X : CommRingCat
f : of PUnit.{u + 1} ⟶ X
x : (forget CommRingCat).obj X
e : f 0 = x
⊢ ({ toMonoidHom := 1, map_zero' := (_ : OneHom.toFun (↑1) 0 = OneHom.toFun (↑1) 0),
          map_add' := (_ : ∀ (x x_1 : ↑X), OneHom.toFun (↑1) (x + x_1) = OneHom.toFun (↑1) (x + x_1)) } ≫
        f)
      x =
    (𝟙 X) x

Next tactic:
exact e
theorem subsingleton_of_isTerminal {X : CommRingCat} (hX : IsTerminal X) : Subsingleton X :=
  (hX.uniqueUpToIso punitIsTerminal).commRingCatIsoToRingEquiv.toEquiv.subsingleton_congr.mpr
    (show Subsingleton PUnit by 

Proof state:
X : CommRingCat
hX : IsTerminal X
⊢ Subsingleton PUnit.{u_1 + 1}

Next tactic:
infer_instance
/-- The product in `CommRingCat` is the cartesian product. -/
def prodFanIsLimit : IsLimit (prodFan A B) where
  lift c := RingHom.prod (c.π.app ⟨WalkingPair.left⟩) (c.π.app ⟨WalkingPair.right⟩)
  fac c j := by
    

Proof state:
A B : CommRingCat
c : Cone (pair A B)
j : Discrete WalkingPair
⊢ (fun c => RingHom.prod (c.π.app { as := WalkingPair.left }) (c.π.app { as := WalkingPair.right })) c ≫
      (prodFan A B).π.app j =
    c.π.app j

Next tactic:
ext
/-- The product in `CommRingCat` is the cartesian product. -/
def prodFanIsLimit : IsLimit (prodFan A B) where
  lift c := RingHom.prod (c.π.app ⟨WalkingPair.left⟩) (c.π.app ⟨WalkingPair.right⟩)
  fac c j := by
    ext
    

Proof state:
case w
A B : CommRingCat
c : Cone (pair A B)
j : Discrete WalkingPair
x✝ : (forget CommRingCat).obj c.pt
⊢ ((fun c => RingHom.prod (c.π.app { as := WalkingPair.left }) (c.π.app { as := WalkingPair.right })) c ≫
        (prodFan A B).π.app j)
      x✝ =
    (c.π.app j) x✝

Next tactic:
rcases j with ⟨⟨⟩⟩
/-- The product in `CommRingCat` is the cartesian product. -/
def prodFanIsLimit : IsLimit (prodFan A B) where
  lift c := RingHom.prod (c.π.app ⟨WalkingPair.left⟩) (c.π.app ⟨WalkingPair.right⟩)
  fac c j := by
    ext
    rcases j with ⟨⟨⟩⟩ <;>
    

Proof state:
case w.mk.left
A B : CommRingCat
c : Cone (pair A B)
x✝ : (forget CommRingCat).obj c.pt
⊢ ((fun c => RingHom.prod (c.π.app { as := WalkingPair.left }) (c.π.app { as := WalkingPair.right })) c ≫
        (prodFan A B).π.app { as := WalkingPair.left })
      x✝ =
    (c.π.app { as := WalkingPair.left }) x✝

Next tactic:
simp only [pair_obj_left, prodFan_pt, BinaryFan.π_app_left, BinaryFan.π_app_right,
      FunctorToTypes.map_comp_apply, forget_map, coe_of, RingHom.prod_apply]
/-- The product in `CommRingCat` is the cartesian product. -/
def prodFanIsLimit : IsLimit (prodFan A B) where
  lift c := RingHom.prod (c.π.app ⟨WalkingPair.left⟩) (c.π.app ⟨WalkingPair.right⟩)
  fac c j := by
    ext
    rcases j with ⟨⟨⟩⟩ <;>
    

Proof state:
case w.mk.right
A B : CommRingCat
c : Cone (pair A B)
x✝ : (forget CommRingCat).obj c.pt
⊢ ((fun c => RingHom.prod (c.π.app { as := WalkingPair.left }) (c.π.app { as := WalkingPair.right })) c ≫
        (prodFan A B).π.app { as := WalkingPair.right })
      x✝ =
    (c.π.app { as := WalkingPair.right }) x✝

Next tactic:
simp only [pair_obj_left, prodFan_pt, BinaryFan.π_app_left, BinaryFan.π_app_right,
      FunctorToTypes.map_comp_apply, forget_map, coe_of, RingHom.prod_apply]
/-- The product in `CommRingCat` is the cartesian product. -/
def prodFanIsLimit : IsLimit (prodFan A B) where
  lift c := RingHom.prod (c.π.app ⟨WalkingPair.left⟩) (c.π.app ⟨WalkingPair.right⟩)
  fac c j := by
    ext
    rcases j with ⟨⟨⟩⟩ <;>
    simp only [pair_obj_left, prodFan_pt, BinaryFan.π_app_left, BinaryFan.π_app_right,
      FunctorToTypes.map_comp_apply, forget_map, coe_of, RingHom.prod_apply] <;>
    

Proof state:
case w.mk.left
A B : CommRingCat
c : Cone (pair A B)
x✝ : (forget CommRingCat).obj c.pt
⊢ (RingHom.prod (BinaryFan.fst c) (BinaryFan.snd c) ≫ BinaryFan.fst (prodFan A B)) x✝ = (BinaryFan.fst c) x✝

Next tactic:
rfl
/-- The product in `CommRingCat` is the cartesian product. -/
def prodFanIsLimit : IsLimit (prodFan A B) where
  lift c := RingHom.prod (c.π.app ⟨WalkingPair.left⟩) (c.π.app ⟨WalkingPair.right⟩)
  fac c j := by
    ext
    rcases j with ⟨⟨⟩⟩ <;>
    simp only [pair_obj_left, prodFan_pt, BinaryFan.π_app_left, BinaryFan.π_app_right,
      FunctorToTypes.map_comp_apply, forget_map, coe_of, RingHom.prod_apply] <;>
    

Proof state:
case w.mk.right
A B : CommRingCat
c : Cone (pair A B)
x✝ : (forget CommRingCat).obj c.pt
⊢ (RingHom.prod (BinaryFan.fst c) (BinaryFan.snd c) ≫ BinaryFan.snd (prodFan A B)) x✝ = (BinaryFan.snd c) x✝

Next tactic:
rfl
/-- The product in `CommRingCat` is the cartesian product. -/
def prodFanIsLimit : IsLimit (prodFan A B) where
  lift c := RingHom.prod (c.π.app ⟨WalkingPair.left⟩) (c.π.app ⟨WalkingPair.right⟩)
  fac c j := by
    ext
    rcases j with ⟨⟨⟩⟩ <;>
    simp only [pair_obj_left, prodFan_pt, BinaryFan.π_app_left, BinaryFan.π_app_right,
      FunctorToTypes.map_comp_apply, forget_map, coe_of, RingHom.prod_apply] <;>
    rfl
  uniq s m h := by
    

Proof state:
A B : CommRingCat
s : Cone (pair A B)
m : s.pt ⟶ (prodFan A B).pt
h : ∀ (j : Discrete WalkingPair), m ≫ (prodFan A B).π.app j = s.π.app j
⊢ m = (fun c => RingHom.prod (c.π.app { as := WalkingPair.left }) (c.π.app { as := WalkingPair.right })) s

Next tactic:
ext x
/-- The product in `CommRingCat` is the cartesian product. -/
def prodFanIsLimit : IsLimit (prodFan A B) where
  lift c := RingHom.prod (c.π.app ⟨WalkingPair.left⟩) (c.π.app ⟨WalkingPair.right⟩)
  fac c j := by
    ext
    rcases j with ⟨⟨⟩⟩ <;>
    simp only [pair_obj_left, prodFan_pt, BinaryFan.π_app_left, BinaryFan.π_app_right,
      FunctorToTypes.map_comp_apply, forget_map, coe_of, RingHom.prod_apply] <;>
    rfl
  uniq s m h := by
    ext x
    

Proof state:
case w
A B : CommRingCat
s : Cone (pair A B)
m : s.pt ⟶ (prodFan A B).pt
h : ∀ (j : Discrete WalkingPair), m ≫ (prodFan A B).π.app j = s.π.app j
x : (forget CommRingCat).obj s.pt
⊢ m x = ((fun c => RingHom.prod (c.π.app { as := WalkingPair.left }) (c.π.app { as := WalkingPair.right })) s) x

Next tactic:
change m x = (BinaryFan.fst s x, BinaryFan.snd s x)
/-- The product in `CommRingCat` is the cartesian product. -/
def prodFanIsLimit : IsLimit (prodFan A B) where
  lift c := RingHom.prod (c.π.app ⟨WalkingPair.left⟩) (c.π.app ⟨WalkingPair.right⟩)
  fac c j := by
    ext
    rcases j with ⟨⟨⟩⟩ <;>
    simp only [pair_obj_left, prodFan_pt, BinaryFan.π_app_left, BinaryFan.π_app_right,
      FunctorToTypes.map_comp_apply, forget_map, coe_of, RingHom.prod_apply] <;>
    rfl
  uniq s m h := by
    ext x
    change m x = (BinaryFan.fst s x, BinaryFan.snd s x)
    

Proof state:
case w
A B : CommRingCat
s : Cone (pair A B)
m : s.pt ⟶ (prodFan A B).pt
h : ∀ (j : Discrete WalkingPair), m ≫ (prodFan A B).π.app j = s.π.app j
x : (forget CommRingCat).obj s.pt
⊢ m x = ((BinaryFan.fst s) x, (BinaryFan.snd s) x)

Next tactic:
have eq1 := congr_hom (h ⟨WalkingPair.left⟩) x