l3lab/ntp-mathlib huggingface dataset:
/-- Witness that the limit cone in `AlgebraCat R` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ AlgebraCatMax.{v, w} R) : IsLimit (limitCone.{v, w} F) := by
  refine'
    IsLimit.ofFaithful (forget (AlgebraCat R)) (Types.limitConeIsLimit.{v, w} _)
      -- Porting note: in mathlib3 the function term
      -- `fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v`
      -- was provided by unification, and the last argument `(fun s => _)` was `(fun s => rfl)`.
      (fun s => ⟨⟨⟨⟨fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v, _⟩,
         _⟩, _⟩, _, _⟩, _⟩)
      (fun s => _)
  · intro j j' f
    exact FunLike.congr_fun (Cone.w s f) v
  · -- Porting note: we could add a custom `ext` lemma here.
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_one, Functor.mapCone_π_app]
    

Proof state:
case refine'_2.a.h
R : Type u
inst✝¹ : CommRing R
J : Type v
inst✝ : SmallCategory J
F : J ⥤ AlgebraCatMax R
s : Cone F
j : J
⊢ (s.π.app j) 1 = ↑1 j

Next tactic:
erw [map_one]
/-- Witness that the limit cone in `AlgebraCat R` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ AlgebraCatMax.{v, w} R) : IsLimit (limitCone.{v, w} F) := by
  refine'
    IsLimit.ofFaithful (forget (AlgebraCat R)) (Types.limitConeIsLimit.{v, w} _)
      -- Porting note: in mathlib3 the function term
      -- `fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v`
      -- was provided by unification, and the last argument `(fun s => _)` was `(fun s => rfl)`.
      (fun s => ⟨⟨⟨⟨fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v, _⟩,
         _⟩, _⟩, _, _⟩, _⟩)
      (fun s => _)
  · intro j j' f
    exact FunLike.congr_fun (Cone.w s f) v
  · -- Porting note: we could add a custom `ext` lemma here.
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_one, Functor.mapCone_π_app]
    erw [map_one]
    

Proof state:
case refine'_2.a.h
R : Type u
inst✝¹ : CommRing R
J : Type v
inst✝ : SmallCategory J
F : J ⥤ AlgebraCatMax R
s : Cone F
j : J
⊢ 1 = ↑1 j

Next tactic:
rfl
/-- Witness that the limit cone in `AlgebraCat R` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ AlgebraCatMax.{v, w} R) : IsLimit (limitCone.{v, w} F) := by
  refine'
    IsLimit.ofFaithful (forget (AlgebraCat R)) (Types.limitConeIsLimit.{v, w} _)
      -- Porting note: in mathlib3 the function term
      -- `fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v`
      -- was provided by unification, and the last argument `(fun s => _)` was `(fun s => rfl)`.
      (fun s => ⟨⟨⟨⟨fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v, _⟩,
         _⟩, _⟩, _, _⟩, _⟩)
      (fun s => _)
  · intro j j' f
    exact FunLike.congr_fun (Cone.w s f) v
  · -- Porting note: we could add a custom `ext` lemma here.
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_one, Functor.mapCone_π_app]
    erw [map_one]
    rfl
  · 

Proof state:
case refine'_3
R : Type u
inst✝¹ : CommRing R
J : Type v
inst✝ : SmallCategory J
F : J ⥤ AlgebraCatMax R
s : Cone F
⊢ ∀ (x y : ↑s.pt),
    OneHom.toFun
        {
          toFun := fun v =>
            { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
              property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) },
          map_one' :=
            (_ :
              (fun v =>
                    { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                      property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) })
                  1 =
                1) }
        (x * y) =
      OneHom.toFun
          {
            toFun := fun v =>
              { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) },
            map_one' :=
              (_ :
                (fun v =>
                      { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                        property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) })
                    1 =
                  1) }
          x *
        OneHom.toFun
          {
            toFun := fun v =>
              { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) },
            map_one' :=
              (_ :
                (fun v =>
                      { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                        property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) })
                    1 =
                  1) }
          y

Next tactic:
intro x y
/-- Witness that the limit cone in `AlgebraCat R` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ AlgebraCatMax.{v, w} R) : IsLimit (limitCone.{v, w} F) := by
  refine'
    IsLimit.ofFaithful (forget (AlgebraCat R)) (Types.limitConeIsLimit.{v, w} _)
      -- Porting note: in mathlib3 the function term
      -- `fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v`
      -- was provided by unification, and the last argument `(fun s => _)` was `(fun s => rfl)`.
      (fun s => ⟨⟨⟨⟨fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v, _⟩,
         _⟩, _⟩, _, _⟩, _⟩)
      (fun s => _)
  · intro j j' f
    exact FunLike.congr_fun (Cone.w s f) v
  · -- Porting note: we could add a custom `ext` lemma here.
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_one, Functor.mapCone_π_app]
    erw [map_one]
    rfl
  · intro x y
    

Proof state:
case refine'_3
R : Type u
inst✝¹ : CommRing R
J : Type v
inst✝ : SmallCategory J
F : J ⥤ AlgebraCatMax R
s : Cone F
x y : ↑s.pt
⊢ OneHom.toFun
      {
        toFun := fun v =>
          { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
            property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) },
        map_one' :=
          (_ :
            (fun v =>
                  { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                    property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) })
                1 =
              1) }
      (x * y) =
    OneHom.toFun
        {
          toFun := fun v =>
            { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
              property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) },
          map_one' :=
            (_ :
              (fun v =>
                    { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                      property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) })
                  1 =
                1) }
        x *
      OneHom.toFun
        {
          toFun := fun v =>
            { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
              property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) },
          map_one' :=
            (_ :
              (fun v =>
                    { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                      property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) })
                  1 =
                1) }
        y

Next tactic:
apply Subtype.ext
/-- Witness that the limit cone in `AlgebraCat R` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ AlgebraCatMax.{v, w} R) : IsLimit (limitCone.{v, w} F) := by
  refine'
    IsLimit.ofFaithful (forget (AlgebraCat R)) (Types.limitConeIsLimit.{v, w} _)
      -- Porting note: in mathlib3 the function term
      -- `fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v`
      -- was provided by unification, and the last argument `(fun s => _)` was `(fun s => rfl)`.
      (fun s => ⟨⟨⟨⟨fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v, _⟩,
         _⟩, _⟩, _, _⟩, _⟩)
      (fun s => _)
  · intro j j' f
    exact FunLike.congr_fun (Cone.w s f) v
  · -- Porting note: we could add a custom `ext` lemma here.
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_one, Functor.mapCone_π_app]
    erw [map_one]
    rfl
  · intro x y
    apply Subtype.ext
    

Proof state:
case refine'_3.a
R : Type u
inst✝¹ : CommRing R
J : Type v
inst✝ : SmallCategory J
F : J ⥤ AlgebraCatMax R
s : Cone F
x y : ↑s.pt
⊢ ↑(OneHom.toFun
        {
          toFun := fun v =>
            { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
              property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) },
          map_one' :=
            (_ :
              (fun v =>
                    { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                      property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) })
                  1 =
                1) }
        (x * y)) =
    ↑(OneHom.toFun
          {
            toFun := fun v =>
              { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) },
            map_one' :=
              (_ :
                (fun v =>
                      { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                        property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) })
                    1 =
                  1) }
          x *
        OneHom.toFun
          {
            toFun := fun v =>
              { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) },
            map_one' :=
              (_ :
                (fun v =>
                      { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                        property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) })
                    1 =
                  1) }
          y)

Next tactic:
ext j
/-- Witness that the limit cone in `AlgebraCat R` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ AlgebraCatMax.{v, w} R) : IsLimit (limitCone.{v, w} F) := by
  refine'
    IsLimit.ofFaithful (forget (AlgebraCat R)) (Types.limitConeIsLimit.{v, w} _)
      -- Porting note: in mathlib3 the function term
      -- `fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v`
      -- was provided by unification, and the last argument `(fun s => _)` was `(fun s => rfl)`.
      (fun s => ⟨⟨⟨⟨fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v, _⟩,
         _⟩, _⟩, _, _⟩, _⟩)
      (fun s => _)
  · intro j j' f
    exact FunLike.congr_fun (Cone.w s f) v
  · -- Porting note: we could add a custom `ext` lemma here.
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_one, Functor.mapCone_π_app]
    erw [map_one]
    rfl
  · intro x y
    apply Subtype.ext
    ext j
    

Proof state:
case refine'_3.a.h
R : Type u
inst✝¹ : CommRing R
J : Type v
inst✝ : SmallCategory J
F : J ⥤ AlgebraCatMax R
s : Cone F
x y : ↑s.pt
j : J
⊢ ↑(OneHom.toFun
          {
            toFun := fun v =>
              { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) },
            map_one' :=
              (_ :
                (fun v =>
                      { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                        property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) })
                    1 =
                  1) }
          (x * y))
      j =
    ↑(OneHom.toFun
            {
              toFun := fun v =>
                { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                  property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) },
              map_one' :=
                (_ :
                  (fun v =>
                        { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                          property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) })
                      1 =
                    1) }
            x *
          OneHom.toFun
            {
              toFun := fun v =>
                { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                  property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) },
              map_one' :=
                (_ :
                  (fun v =>
                        { val := fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v,
                          property := (_ : ∀ {j j' : J} (f : j ⟶ j'), (s.π.app j ≫ F.map f) v = (s.π.app j') v) })
                      1 =
                    1) }
            y)
      j

Next tactic:
simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
/-- Witness that the limit cone in `AlgebraCat R` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ AlgebraCatMax.{v, w} R) : IsLimit (limitCone.{v, w} F) := by
  refine'
    IsLimit.ofFaithful (forget (AlgebraCat R)) (Types.limitConeIsLimit.{v, w} _)
      -- Porting note: in mathlib3 the function term
      -- `fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v`
      -- was provided by unification, and the last argument `(fun s => _)` was `(fun s => rfl)`.
      (fun s => ⟨⟨⟨⟨fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v, _⟩,
         _⟩, _⟩, _, _⟩, _⟩)
      (fun s => _)
  · intro j j' f
    exact FunLike.congr_fun (Cone.w s f) v
  · -- Porting note: we could add a custom `ext` lemma here.
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_one, Functor.mapCone_π_app]
    erw [map_one]
    rfl
  · intro x y
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below, but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_mul, Functor.mapCone_π_app]
    

Proof state:
case refine'_3.a.h
R : Type u
inst✝¹ : CommRing R
J : Type v
inst✝ : SmallCategory J
F : J ⥤ AlgebraCatMax R
s : Cone F
x y : ↑s.pt
j : J
⊢ (s.π.app j) (x * y) =
    ↑({ val := fun j => (s.π.app j) x,
            property := (_ : (fun x => x ∈ Functor.sections (F ⋙ forget (AlgebraCatMax R))) fun j => (s.π.app j) x) } *
          { val := fun j => (s.π.app j) y,
            property := (_ : (fun x => x ∈ Functor.sections (F ⋙ forget (AlgebraCatMax R))) fun j => (s.π.app j) y) })
      j

Next tactic:
erw [map_mul]
/-- Witness that the limit cone in `AlgebraCat R` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ AlgebraCatMax.{v, w} R) : IsLimit (limitCone.{v, w} F) := by
  refine'
    IsLimit.ofFaithful (forget (AlgebraCat R)) (Types.limitConeIsLimit.{v, w} _)
      -- Porting note: in mathlib3 the function term
      -- `fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v`
      -- was provided by unification, and the last argument `(fun s => _)` was `(fun s => rfl)`.
      (fun s => ⟨⟨⟨⟨fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v, _⟩,
         _⟩, _⟩, _, _⟩, _⟩)
      (fun s => _)
  · intro j j' f
    exact FunLike.congr_fun (Cone.w s f) v
  · -- Porting note: we could add a custom `ext` lemma here.
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_one, Functor.mapCone_π_app]
    erw [map_one]
    rfl
  · intro x y
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below, but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_mul, Functor.mapCone_π_app]
    erw [map_mul]
    

Proof state:
case refine'_3.a.h
R : Type u
inst✝¹ : CommRing R
J : Type v
inst✝ : SmallCategory J
F : J ⥤ AlgebraCatMax R
s : Cone F
x y : ↑s.pt
j : J
⊢ (s.π.app j) x * (s.π.app j) y =
    ↑({ val := fun j => (s.π.app j) x,
            property := (_ : (fun x => x ∈ Functor.sections (F ⋙ forget (AlgebraCatMax R))) fun j => (s.π.app j) x) } *
          { val := fun j => (s.π.app j) y,
            property := (_ : (fun x => x ∈ Functor.sections (F ⋙ forget (AlgebraCatMax R))) fun j => (s.π.app j) y) })
      j

Next tactic:
rfl
/-- Witness that the limit cone in `AlgebraCat R` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ AlgebraCatMax.{v, w} R) : IsLimit (limitCone.{v, w} F) := by
  refine'
    IsLimit.ofFaithful (forget (AlgebraCat R)) (Types.limitConeIsLimit.{v, w} _)
      -- Porting note: in mathlib3 the function term
      -- `fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v`
      -- was provided by unification, and the last argument `(fun s => _)` was `(fun s => rfl)`.
      (fun s => ⟨⟨⟨⟨fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v, _⟩,
         _⟩, _⟩, _, _⟩, _⟩)
      (fun s => _)
  · intro j j' f
    exact FunLike.congr_fun (Cone.w s f) v
  · -- Porting note: we could add a custom `ext` lemma here.
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_one, Functor.mapCone_π_app]
    erw [map_one]
    rfl
  · intro x y
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below, but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_mul, Functor.mapCone_π_app]
    erw [map_mul]
    rfl
  · simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- The below `simp` was enough before leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_zero, Functor.mapCone_π_app]
    

Proof state:
case refine'_4
R : Type u
inst✝¹ : CommRing R
J : Type v
inst✝ : SmallCategory J
F : J ⥤ AlgebraCatMax R
s : Cone F
⊢ { val := fun j => (s.π.app j) 0,
      property := (_ : (fun x => x ∈ Functor.sections (F ⋙ forget (AlgebraCatMax R))) fun j => (s.π.app j) 0) } =
    0

Next tactic:
apply Subtype.ext
/-- Witness that the limit cone in `AlgebraCat R` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ AlgebraCatMax.{v, w} R) : IsLimit (limitCone.{v, w} F) := by
  refine'
    IsLimit.ofFaithful (forget (AlgebraCat R)) (Types.limitConeIsLimit.{v, w} _)
      -- Porting note: in mathlib3 the function term
      -- `fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v`
      -- was provided by unification, and the last argument `(fun s => _)` was `(fun s => rfl)`.
      (fun s => ⟨⟨⟨⟨fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v, _⟩,
         _⟩, _⟩, _, _⟩, _⟩)
      (fun s => _)
  · intro j j' f
    exact FunLike.congr_fun (Cone.w s f) v
  · -- Porting note: we could add a custom `ext` lemma here.
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_one, Functor.mapCone_π_app]
    erw [map_one]
    rfl
  · intro x y
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below, but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_mul, Functor.mapCone_π_app]
    erw [map_mul]
    rfl
  · simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- The below `simp` was enough before leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_zero, Functor.mapCone_π_app]
    apply Subtype.ext
    

Proof state:
case refine'_4.a
R : Type u
inst✝¹ : CommRing R
J : Type v
inst✝ : SmallCategory J
F : J ⥤ AlgebraCatMax R
s : Cone F
⊢ ↑{ val := fun j => (s.π.app j) 0,
        property := (_ : (fun x => x ∈ Functor.sections (F ⋙ forget (AlgebraCatMax R))) fun j => (s.π.app j) 0) } =
    ↑0

Next tactic:
dsimp
/-- Witness that the limit cone in `AlgebraCat R` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit (F : J ⥤ AlgebraCatMax.{v, w} R) : IsLimit (limitCone.{v, w} F) := by
  refine'
    IsLimit.ofFaithful (forget (AlgebraCat R)) (Types.limitConeIsLimit.{v, w} _)
      -- Porting note: in mathlib3 the function term
      -- `fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v`
      -- was provided by unification, and the last argument `(fun s => _)` was `(fun s => rfl)`.
      (fun s => ⟨⟨⟨⟨fun v => ⟨fun j => ((forget (AlgebraCat R)).mapCone s).π.app j v, _⟩,
         _⟩, _⟩, _, _⟩, _⟩)
      (fun s => _)
  · intro j j' f
    exact FunLike.congr_fun (Cone.w s f) v
  · -- Porting note: we could add a custom `ext` lemma here.
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_one, Functor.mapCone_π_app]
    erw [map_one]
    rfl
  · intro x y
    apply Subtype.ext
    ext j
    simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- This used to be as below, but we need `erw` after leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_mul, Functor.mapCone_π_app]
    erw [map_mul]
    rfl
  · simp only [Functor.comp_obj, Functor.mapCone_pt, Functor.mapCone_π_app,
      forget_map_eq_coe]
    -- The below `simp` was enough before leanprover/lean4#2644
    -- simp [forget_map_eq_coe, AlgHom.map_zero, Functor.mapCone_π_app]
    apply Subtype.ext
    dsimp
    

Proof state:
case refine'_4.a
R : Type u
inst✝¹ : CommRing R
J : Type v
inst✝ : SmallCategory J
F : J ⥤ AlgebraCatMax R
s : Cone F
⊢ (fun j => (s.π.app j) 0) = ↑0

Next tactic:
funext u