awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : Cone F
h :
  ∀ s ∈ F.sections,
    ∃! x : c.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          s j
this :
  ∀ (c_1 : Cone F) (y : c_1.pt),
    ∃! x : c.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          (↑(sectionOfCone c_1 y) : (j : J) → F.obj j) j
refine_2 :
  (x : (c_1 : Cone F) → (y : c_1.pt) → c.pt) →
    (hx :
        ∀ (c_1 : Cone F) (y : c_1.pt),
          (fun (x : c.pt) ↦
                ∀ (j : J),
                  (hom (FC := TypeCat.Fun) (c.π.app j) :
                        (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) →
                          (fun (X : Type u) ↦ X) (F.obj j))
                      x =
                    (↑(sectionOfCone c_1 y) : (j : J) → F.obj j) j)
              (x c_1 y) ∧
            ∀ (y_1 : c.pt),
              (fun (x : c.pt) ↦
                    ∀ (j : J),
                      (hom (FC := TypeCat.Fun) (c.π.app j) :
                            (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) →
                              (fun (X : Type u) ↦ X) (F.obj j))
                          x =
                        (↑(sectionOfCone c_1 y) : (j : J) → F.obj j) j)
                  y_1 →
                y_1 = x c_1 y) →
      IsLimit c
⊢ IsLimit c

Tactic:
choose x hx using fun c y ↦ h _ (sectionOfCone c y).2

Premises used: Subtype.val, Subtype.property, CategoryTheory.Limits.Types.sectionOfCone
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c_1 : Cone F
h :
  ∀ s ∈ F.sections,
    ∃! x : c_1.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c_1.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          s j
this :
  ∀ (c : Cone F) (y : c.pt),
    ∃! x : c_1.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c_1.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          (↑(sectionOfCone c y) : (j : J) → F.obj j) j
x : (c : Cone F) → (y : c.pt) → c_1.pt
hx :
  ∀ (c : Cone F) (y : c.pt),
    (fun (x : c_1.pt) ↦
          ∀ (j : J),
            (hom (FC := TypeCat.Fun) (c_1.π.app j) :
                  (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) →
                    (fun (X : Type u) ↦ X) (F.obj j))
                x =
              (↑(sectionOfCone c y) : (j : J) → F.obj j) j)
        (x c y) ∧
      ∀ (y_1 : c_1.pt),
        (fun (x : c_1.pt) ↦
              ∀ (j : J),
                (hom (FC := TypeCat.Fun) (c_1.π.app j) :
                      (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) →
                        (fun (X : Type u) ↦ X) (F.obj j))
                    x =
                  (↑(sectionOfCone c y) : (j : J) → F.obj j) j)
            y_1 →
          y_1 = x c y
c : Cone F
j : J
h₁ :
  ∀ (y : c.pt),
    TypeCat.Fun.toFun (hom (FC := TypeCat.Fun) ((fun (d : Cone F) ↦ ↾x d) c ≫ c_1.π.app j)) y =
      (hom (FC := TypeCat.Fun) (c.π.app j)).toFun y
⊢ (fun (d : Cone F) ↦ ↾x d) c ≫ c_1.π.app j = c.π.app j

Tactic:
ext y
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c_1 : Cone F
h :
  ∀ s ∈ F.sections,
    ∃! x : c_1.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c_1.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          s j
this :
  ∀ (c : Cone F) (y : c.pt),
    ∃! x : c_1.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c_1.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          (↑(sectionOfCone c y) : (j : J) → F.obj j) j
x : (c : Cone F) → (y : c.pt) → c_1.pt
hx :
  ∀ (c : Cone F) (y : c.pt),
    (fun (x : c_1.pt) ↦
          ∀ (j : J),
            (hom (FC := TypeCat.Fun) (c_1.π.app j) :
                  (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) →
                    (fun (X : Type u) ↦ X) (F.obj j))
                x =
              (↑(sectionOfCone c y) : (j : J) → F.obj j) j)
        (x c y) ∧
      ∀ (y_1 : c_1.pt),
        (fun (x : c_1.pt) ↦
              ∀ (j : J),
                (hom (FC := TypeCat.Fun) (c_1.π.app j) :
                      (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) →
                        (fun (X : Type u) ↦ X) (F.obj j))
                    x =
                  (↑(sectionOfCone c y) : (j : J) → F.obj j) j)
            y_1 →
          y_1 = x c y
c : Cone F
j : J
y : c.pt
⊢ TypeCat.Fun.toFun (hom (FC := TypeCat.Fun) ((fun (d : Cone F) ↦ ↾x d) c ≫ c_1.π.app j)) y =
    (hom (FC := TypeCat.Fun) (c.π.app j)).toFun y

Tactic:
(hx c y).1 j

Premises used: And.left
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c_1 : Cone F
h :
  ∀ s ∈ F.sections,
    ∃! x : c_1.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c_1.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          s j
this :
  ∀ (c : Cone F) (y : c.pt),
    ∃! x : c_1.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c_1.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          (↑(sectionOfCone c y) : (j : J) → F.obj j) j
x : (c : Cone F) → (y : c.pt) → c_1.pt
hx :
  ∀ (c : Cone F) (y : c.pt),
    (fun (x : c_1.pt) ↦
          ∀ (j : J),
            (hom (FC := TypeCat.Fun) (c_1.π.app j) :
                  (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) →
                    (fun (X : Type u) ↦ X) (F.obj j))
                x =
              (↑(sectionOfCone c y) : (j : J) → F.obj j) j)
        (x c y) ∧
      ∀ (y_1 : c_1.pt),
        (fun (x : c_1.pt) ↦
              ∀ (j : J),
                (hom (FC := TypeCat.Fun) (c_1.π.app j) :
                      (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) →
                        (fun (X : Type u) ↦ X) (F.obj j))
                    x =
                  (↑(sectionOfCone c y) : (j : J) → F.obj j) j)
            y_1 →
          y_1 = x c y
c : Cone F
j : J
y : c.pt
⊢ TypeCat.Fun.toFun (hom (FC := TypeCat.Fun) ((fun (d : Cone F) ↦ ↾x d) c ≫ c_1.π.app j)) y =
    (hom (FC := TypeCat.Fun) (c.π.app j)).toFun y

Tactic:
exact (hx c y).1 j

Premises used: And.left
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c_1 : Cone F
h :
  ∀ s ∈ F.sections,
    ∃! x : c_1.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c_1.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          s j
this :
  ∀ (c : Cone F) (y : c.pt),
    ∃! x : c_1.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c_1.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          (↑(sectionOfCone c y) : (j : J) → F.obj j) j
x : (c : Cone F) → (y : c.pt) → c_1.pt
hx :
  ∀ (c : Cone F) (y : c.pt),
    (fun (x : c_1.pt) ↦
          ∀ (j : J),
            (hom (FC := TypeCat.Fun) (c_1.π.app j) :
                  (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) →
                    (fun (X : Type u) ↦ X) (F.obj j))
                x =
              (↑(sectionOfCone c y) : (j : J) → F.obj j) j)
        (x c y) ∧
      ∀ (y_1 : c_1.pt),
        (fun (x : c_1.pt) ↦
              ∀ (j : J),
                (hom (FC := TypeCat.Fun) (c_1.π.app j) :
                      (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) →
                        (fun (X : Type u) ↦ X) (F.obj j))
                    x =
                  (↑(sectionOfCone c y) : (j : J) → F.obj j) j)
            y_1 →
          y_1 = x c y
c : Cone F
f : c.pt ⟶ c_1.pt
hf : ∀ (j : J), f ≫ c_1.π.app j = c.π.app j
h₁ :
  ∀ (y : c.pt),
    (hom (FC := TypeCat.Fun) f).toFun y =
      TypeCat.Fun.toFun (@hom _ types TypeCat.Fun _ _ _ _ _ ((fun (d : Cone F) ↦ ↾x d) c)) y
⊢ f = (fun (d : Cone F) ↦ ↾x d) c

Tactic:
ext y
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c_1 : Cone F
h :
  ∀ s ∈ F.sections,
    ∃! x : c_1.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c_1.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          s j
this :
  ∀ (c : Cone F) (y : c.pt),
    ∃! x : c_1.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c_1.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          (↑(sectionOfCone c y) : (j : J) → F.obj j) j
x : (c : Cone F) → (y : c.pt) → c_1.pt
hx :
  ∀ (c : Cone F) (y : c.pt),
    (fun (x : c_1.pt) ↦
          ∀ (j : J),
            (hom (FC := TypeCat.Fun) (c_1.π.app j) :
                  (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) →
                    (fun (X : Type u) ↦ X) (F.obj j))
                x =
              (↑(sectionOfCone c y) : (j : J) → F.obj j) j)
        (x c y) ∧
      ∀ (y_1 : c_1.pt),
        (fun (x : c_1.pt) ↦
              ∀ (j : J),
                (hom (FC := TypeCat.Fun) (c_1.π.app j) :
                      (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) →
                        (fun (X : Type u) ↦ X) (F.obj j))
                    x =
                  (↑(sectionOfCone c y) : (j : J) → F.obj j) j)
            y_1 →
          y_1 = x c y
c : Cone F
f : c.pt ⟶ c_1.pt
hf : ∀ (j : J), f ≫ c_1.π.app j = c.π.app j
y : c.pt
⊢ (hom (FC := TypeCat.Fun) f).toFun y =
    TypeCat.Fun.toFun (@hom _ types TypeCat.Fun _ _ _ _ _ ((fun (d : Cone F) ↦ ↾x d) c)) y

Tactic:
(hx c y).2 (f y) (fun j ↦ congr_hom (hf j) y)

Premises used: CategoryTheory.ConcreteCategory.hom, And.right, DFunLike.coe
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c_1 : Cone F
h :
  ∀ s ∈ F.sections,
    ∃! x : c_1.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c_1.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          s j
this :
  ∀ (c : Cone F) (y : c.pt),
    ∃! x : c_1.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c_1.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          (↑(sectionOfCone c y) : (j : J) → F.obj j) j
x : (c : Cone F) → (y : c.pt) → c_1.pt
hx :
  ∀ (c : Cone F) (y : c.pt),
    (fun (x : c_1.pt) ↦
          ∀ (j : J),
            (hom (FC := TypeCat.Fun) (c_1.π.app j) :
                  (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) →
                    (fun (X : Type u) ↦ X) (F.obj j))
                x =
              (↑(sectionOfCone c y) : (j : J) → F.obj j) j)
        (x c y) ∧
      ∀ (y_1 : c_1.pt),
        (fun (x : c_1.pt) ↦
              ∀ (j : J),
                (hom (FC := TypeCat.Fun) (c_1.π.app j) :
                      (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c_1.pt).obj j)) →
                        (fun (X : Type u) ↦ X) (F.obj j))
                    x =
                  (↑(sectionOfCone c y) : (j : J) → F.obj j) j)
            y_1 →
          y_1 = x c y
c : Cone F
f : c.pt ⟶ c_1.pt
hf : ∀ (j : J), f ≫ c_1.π.app j = c.π.app j
y : c.pt
⊢ (hom (FC := TypeCat.Fun) f).toFun y =
    TypeCat.Fun.toFun (@hom _ types TypeCat.Fun _ _ _ _ _ ((fun (d : Cone F) ↦ ↾x d) c)) y

Tactic:
exact (hx c y).2 (f y) (fun j ↦ congr_hom (hf j) y)

Premises used: CategoryTheory.ConcreteCategory.hom, And.right, DFunLike.coe
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : Cone F
h :
  ∀ s ∈ F.sections,
    ∃! x : c.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          s j
this :
  ∀ (c_1 : Cone F) (y : c_1.pt),
    ∃! x : c.pt,
      ∀ (j : J),
        (hom (FC := TypeCat.Fun) (c.π.app j) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
            x =
          (↑(sectionOfCone c_1 y) : (j : J) → F.obj j) j
x : (c_1 : Cone F) → (y : c_1.pt) → c.pt
hx :
  ∀ (c_1 : Cone F) (y : c_1.pt),
    (fun (x : c.pt) ↦
          ∀ (j : J),
            (hom (FC := TypeCat.Fun) (c.π.app j) :
                  (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
                x =
              (↑(sectionOfCone c_1 y) : (j : J) → F.obj j) j)
        (x c_1 y) ∧
      ∀ (y_1 : c.pt),
        (fun (x : c.pt) ↦
              ∀ (j : J),
                (hom (FC := TypeCat.Fun) (c.π.app j) :
                      (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) →
                        (fun (X : Type u) ↦ X) (F.obj j))
                    x =
                  (↑(sectionOfCone c_1 y) : (j : J) → F.obj j) j)
            y_1 →
          y_1 = x c_1 y
⊢ IsLimit c

Tactic:
exact ⟨fun d ↦ ↾(x d), fun c j ↦ by ext y; exact (hx c y).1 j,
      fun c f hf ↦ by ext y; exact (hx c y).2 (f y) (fun j ↦ congr_hom (hf j) y)⟩
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : Cone F
simp_rw :
  (∀ s ∈ F.sections,
      ∃! x : c.pt,
        ∀ (j : J), (hom (FC := TypeCat.Fun) (c.π.app j) : (a : ((Functor.const J).obj c.pt).obj j) → F.obj j) x = s j) ↔
    Function.Bijective (sectionOfCone c)
simp_rw₁ :
  (∀ s ∈ F.sections,
      ∃! x : c.pt,
        ∀ (j : J), (hom (FC := TypeCat.Fun) (c.π.app j) : (a : ((Functor.const J).obj c.pt).obj j) → F.obj j) x = s j) ↔
    ∀ (b : (↑F.sections : Type (max u v))), ∃! a : c.pt, sectionOfCone c a = b
simp_rw₂ :
  (∀ s ∈ F.sections,
      ∃! x : c.pt,
        ∀ (j : J), (hom (FC := TypeCat.Fun) (c.π.app j) : (a : ((Functor.const J).obj c.pt).obj j) → F.obj j) x = s j) ↔
    ∀ (a : (j : J) → F.obj j) (b : a ∈ F.sections),
      ∃! a_1 : c.pt, sectionOfCone c a_1 = Subtype.mk (p := fun (a : (j : J) → F.obj j) ↦ a ∈ F.sections) a b
simp_rw₃ :
  (∀ s ∈ F.sections,
      ∃! x : c.pt,
        ∀ (j : J), (hom (FC := TypeCat.Fun) (c.π.app j) : (a : ((Functor.const J).obj c.pt).obj j) → F.obj j) x = s j) ↔
    ∀ (a : (j : J) → F.obj j) (b : a ∈ F.sections),
      ∃! a_1 : c.pt, ∀ (j : J), (↑(sectionOfCone c a_1) : (j : J) → F.obj j) j = a j
⊢ Nonempty (IsLimit c) ↔ Function.Bijective (sectionOfCone c)

Tactic:
simp_rw [isLimit_iff, Function.bijective_iff_existsUnique, Subtype.forall, F.sections_ext_iff,
    sectionOfCone]

Premises used: Subtype.forall, CategoryTheory.Functor.sections_ext_iff, Function.bijective_iff_existsUnique, CategoryTheory.Limits.Types.isLimit_iff, CategoryTheory.Limits.Types.sectionOfCone
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : Cone F
t : IsLimit c
j : J
x : c.pt
⊢ Eq.{u + 1} (α := F.obj j)
    ((↑((isLimitEquivSections t : (a : c.pt) → (↑F.sections : Type (max u v))) x) : (j : J) → F.obj j) j : F.obj j)
    ((hom (FC := TypeCat.Fun) (c.π.app j) :
          (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
        x :
      (fun (X : Type u) ↦ X) (F.obj j))

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : Cone F
t : IsLimit c
x : (↑F.sections : Type (max u v))
j : J
rw : (↑((isLimitEquivSections t).toFun ((isLimitEquivSections t).invFun x)) : (j : J) → F.obj j) j
| (↑x : (j : J) → F.obj j) j

Tactic:
rw [← (isLimitEquivSections t).right_inv x]

Premises used: Equiv.right_inv, CategoryTheory.Limits.Types.isLimitEquivSections
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : Cone F
t : IsLimit c
x : (↑F.sections : Type (max u v))
j : J
conv_rhs :
  (hom (FC := TypeCat.Fun) (c.π.app j) : (a : c.pt) → F.obj j)
      (((isLimitEquivSections t).symm : (a : (↑F.sections : Type (max u v))) → c.pt) x : c.pt) =
    (↑((isLimitEquivSections t).toFun ((isLimitEquivSections t).invFun x)) : (j : J) → F.obj j) j
⊢ (hom (FC := TypeCat.Fun) (c.π.app j) : (a : c.pt) → F.obj j)
      (((isLimitEquivSections t).symm : (a : (↑F.sections : Type (max u v))) → c.pt) x : c.pt) =
    (↑x : (j : J) → F.obj j) j

Tactic:
conv_rhs => rw [← (isLimitEquivSections t).right_inv x]

Premises used: Equiv.right_inv, CategoryTheory.Limits.Types.isLimitEquivSections
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
c : Cone F
t : IsLimit c
x : (↑F.sections : Type (max u v))
j : J
⊢ (hom (FC := TypeCat.Fun) (c.π.app j) : (a : c.pt) → F.obj j)
      (((isLimitEquivSections t).symm : (a : (↑F.sections : Type (max u v))) → c.pt) x : c.pt) =
    (↑((isLimitEquivSections t).toFun ((isLimitEquivSections t).invFun x)) : (j : J) → F.obj j) j

Tactic:
rfl
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
inst_1 : Small.{u, max u v} (↑F.sections : Type (max u v))
x y : (limitCone F).pt
w :
  Eq (α := (↑F.sections : Type (max u v)))
    ((Equiv.symm (equivShrink (↑F.sections : Type (max u v))) :
          (a : Shrink.{u, max u v} (↑F.sections : Type (max u v))) → (↑F.sections : Type (max u v)))
        x :
      (↑F.sections : Type (max u v)))
    ((Equiv.symm (equivShrink (↑F.sections : Type (max u v))) :
          (a : Shrink.{u, max u v} (↑F.sections : Type (max u v))) → (↑F.sections : Type (max u v)))
        y :
      (↑F.sections : Type (max u v)))
⊢ x = y

Tactic:
simp_all
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
inst_1 : Small.{u, max u v} (↑F.sections : Type (max u v))
x : Cone F
x_1 : x.pt ⟶ (limitCone F).pt
w : ∀ (j : J), x_1 ≫ (limitCone F).π.app j = x.π.app j
h :
  ∀ (x_2 : x.pt) (j : J),
    Eq.{u + 1} (α := F.obj j)
      ((↑((Equiv.symm (equivShrink (↑F.sections : Type (max u v))) :
                (a : Shrink.{u, max u v} (↑F.sections : Type (max u v))) → (↑F.sections : Type (max u v)))
              ((hom (FC := TypeCat.Fun) x_1).toFun x_2)) :
          (j : J) → F.obj j)
        j)
      ((↑((Equiv.symm (equivShrink (↑F.sections : Type (max u v))) :
                (a : Shrink.{u, max u v} (↑F.sections : Type (max u v))) → (↑F.sections : Type (max u v)))
              (TypeCat.Fun.toFun
                (hom (FC := TypeCat.Fun)
                  (↾fun (v : x.pt) ↦
                    (equivShrink (↑F.sections : Type (max u v)) :
                        (a : (↑F.sections : Type (max u v))) → Shrink.{u, max u v} (↑F.sections : Type (max u v)))
                      (Subtype.mk (p := fun (x : (j : J) → F.obj j) ↦ x ∈ F.sections)
                        (fun (j : J) ↦
                          (hom (FC := TypeCat.Fun) (x.π.app j) :
                              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj x.pt).obj j)) →
                                (fun (X : Type u) ↦ X) (F.obj j))
                            v)
                        ⋯)))
                x_2)) :
          (j : J) → F.obj j)
        j)
⊢ x_1 =
    ↾fun (v : x.pt) ↦
      (equivShrink (↑F.sections : Type (max u v)) :
          (a : (↑F.sections : Type (max u v))) → Shrink.{u, max u v} (↑F.sections : Type (max u v)))
        (Subtype.mk (p := fun (x : (j : J) → F.obj j) ↦ x ∈ F.sections)
          (fun (j : J) ↦
            (hom (FC := TypeCat.Fun) (x.π.app j) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj x.pt).obj j)) → (fun (X : Type u) ↦ X) (F.obj j))
              v)
          ⋯)

Tactic:
ext x j
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
inst_1 : Small.{u, max u v} (↑F.sections : Type (max u v))
x_1 : Cone F
x_2 : x_1.pt ⟶ (limitCone F).pt
w : ∀ (j : J), x_2 ≫ (limitCone F).π.app j = x_1.π.app j
x : x_1.pt
j : J
⊢ Eq.{u + 1} (α := F.obj j)
    ((↑((Equiv.symm (equivShrink (↑F.sections : Type (max u v))) :
              (a : Shrink.{u, max u v} (↑F.sections : Type (max u v))) → (↑F.sections : Type (max u v)))
            ((hom (FC := TypeCat.Fun) x_2).toFun x)) :
        (j : J) → F.obj j)
      j)
    ((↑((Equiv.symm (equivShrink (↑F.sections : Type (max u v))) :
              (a : Shrink.{u, max u v} (↑F.sections : Type (max u v))) → (↑F.sections : Type (max u v)))
            (TypeCat.Fun.toFun
              (hom (FC := TypeCat.Fun)
                (↾fun (v : x_1.pt) ↦
                  (equivShrink (↑F.sections : Type (max u v)) :
                      (a : (↑F.sections : Type (max u v))) → Shrink.{u, max u v} (↑F.sections : Type (max u v)))
                    (Subtype.mk (p := fun (x : (j : J) → F.obj j) ↦ x ∈ F.sections)
                      (fun (j : J) ↦
                        (hom (FC := TypeCat.Fun) (x_1.π.app j) :
                            (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj x_1.pt).obj j)) →
                              (fun (X : Type u) ↦ X) (F.obj j))
                          v)
                      ⋯)))
              x)) :
        (j : J) → F.obj j)
      j)

Tactic:
simpa using congr_hom (w j) x

Premises used: CategoryTheory.ConcreteCategory.congr_hom
-- Module: Mathlib.CategoryTheory.Limits.Types.Limits
Goal:
J : Type v
inst : Category.{w, v} J
F : Functor.{w, u, v, u + 1} J (Type u)
⊢ HasLimit F ↔ Small.{u, max u v} (↑F.sections : Type (max u v))

Tactic:
⟨fun _ => .mk ⟨_, ⟨(Equiv.ofBijective _
    ((isLimit_iff_bijective_sectionOfCone (limit.cone F)).mp ⟨limit.isLimit _⟩)).symm⟩⟩,
   fun _ => ⟨_, Small.limitConeIsLimit F⟩⟩

Premises used: CategoryTheory.Limits.Types.Small.limitCone, CategoryTheory.Limits.limit.cone, Small.mk, CategoryTheory.Limits.Types.Small.limitConeIsLimit, CategoryTheory.Limits.Cone.pt, Nonempty.intro, CategoryTheory.Limits.limit.isLimit, CategoryTheory.Limits.HasLimit.mk', Equiv.ofBijective, Iff.intro, Exists.intro, Iff.mp, CategoryTheory.Limits.LimitCone.mk, Equiv.symm, CategoryTheory.Limits.Types.isLimit_iff_bijective_sectionOfCone, CategoryTheory.Limits.Types.sectionOfCone