awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Sphere (goal)
Goal:
n : ℕ
t : Set (ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
h : IsOpen t
s : Set (↑(Metric.closedBall 0 1) : Type)
r : Set (EuclideanSpace ℝ (Fin n))
hro : IsOpen r
hrs :
  Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
        Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
      r =
    s
hst : ULift.down ⁻¹' s = t
⊢ ∃ (t : Set (↑(Metric.sphere 0 1) : Type)),
    IsOpen t ∧
      ULift.down ⁻¹' t =
        Set.preimage (β := ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
          (fun (x : ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type)) ↦
            match (motive :=
              ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type) → ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type)) x with
            | { down := ⟨p, hp⟩ } =>
              {
                down :=
                  Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                    Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) p ⋯ })
          (ULift.down ⁻¹'
            Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
              r)
have h : IsOpen (X := (↑(Metric.sphere 0 1) : Type))
    (Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
        Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.sphere 0 1) x) ⁻¹'
      r) ∧
  Eq.{u_1 + 1} (α := Set (ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type)))
    (ULift.down ⁻¹'
      Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
          Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.sphere 0 1) x) ⁻¹'
        r)
    (Set.preimage (β := ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
      (fun (x : ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type)) ↦
        match (motive :=
          ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type) → ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type)) x with
        | { down := ⟨p, hp⟩ } =>
          {
            down :=
              Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) p sorry })
      (ULift.down ⁻¹'
        Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
            Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
          r))
Tactic: tauto
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Sphere (goal)
Goal:
n : ℕ
t : Set (ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
h_1 : IsOpen t
s : Set (↑(Metric.closedBall 0 1) : Type)
r : Set (EuclideanSpace ℝ (Fin n))
hro : IsOpen r
hrs :
  Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
        Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
      r =
    s
hst : ULift.down ⁻¹' s = t
h :
  IsOpen (X := (↑(Metric.sphere 0 1) : Type))
      (Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
          Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.sphere 0 1) x) ⁻¹'
        r) ∧
    Eq.{u_1 + 1} (α := Set (ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type)))
      (ULift.down ⁻¹'
        Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
            Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.sphere 0 1) x) ⁻¹'
          r)
      (Set.preimage (β := ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
        (fun (x : ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type)) ↦
          match (motive :=
            ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type) → ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type)) x with
          | { down := ⟨p, hp⟩ } =>
            {
              down :=
                Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                  Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) p ⋯ })
        (ULift.down ⁻¹'
          Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
              Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
            r))
⊢ ∃ (t : Set (↑(Metric.sphere 0 1) : Type)),
    IsOpen t ∧
      ULift.down ⁻¹' t =
        Set.preimage (β := ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
          (fun (x : ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type)) ↦
            match (motive :=
              ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type) → ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type)) x with
            | { down := ⟨p, hp⟩ } =>
              {
                down :=
                  Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                    Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) p ⋯ })
          (ULift.down ⁻¹'
            Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
              r)
have w : Set (↑(Metric.sphere 0 1) : Type)
Tactic: tauto
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Sphere (goal)
Goal:
n : ℕ
t : Set (ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
h : IsOpen t
s : Set (↑(Metric.closedBall 0 1) : Type)
r : Set (EuclideanSpace ℝ (Fin n))
hro : IsOpen r
hrs :
  Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
        Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
      r =
    s
hst : ULift.down ⁻¹' s = t
⊢ IsOpen (X := ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type))
    ((fun (x : ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type)) ↦
        match (motive := ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type) → ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
          x with
        | { down := ⟨p, hp⟩ } =>
          {
            down :=
              Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) p ⋯ }) ⁻¹'
      t)
have rw : ∃ (t : Set (↑(Metric.ball 0 1) : Type)),
  IsOpen t ∧
    ULift.down ⁻¹' t =
      Set.preimage (β := ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
        (fun (x : ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type)) ↦
          match (motive :=
            ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type) → ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type)) x with
          | { down := ⟨p, hp⟩ } =>
            {
              down :=
                Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                  Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) p sorry })
        (ULift.down ⁻¹'
          Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
              Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
            r)
Tactic: rw [isOpen_induced_iff, ← hst, ← hrs]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Sphere (goal)
Goal:
n : ℕ
t : Set (ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
h : IsOpen t
s : Set (↑(Metric.closedBall 0 1) : Type)
r : Set (EuclideanSpace ℝ (Fin n))
hro : IsOpen r
hrs :
  Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
        Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
      r =
    s
hst : ULift.down ⁻¹' s = t
⊢ ∃ (t : Set (↑(Metric.ball 0 1) : Type)),
    IsOpen t ∧
      ULift.down ⁻¹' t =
        Set.preimage (β := ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
          (fun (x : ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type)) ↦
            match (motive :=
              ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type) → ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type)) x with
            | { down := ⟨p, hp⟩ } =>
              {
                down :=
                  Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                    Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) p ⋯ })
          (ULift.down ⁻¹'
            Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
              r)
have h : IsOpen (X := (↑(Metric.ball 0 1) : Type))
    (Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
        Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.ball 0 1) x) ⁻¹'
      r) ∧
  Eq.{u_1 + 1} (α := Set (ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type)))
    (ULift.down ⁻¹'
      Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
          Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.ball 0 1) x) ⁻¹'
        r)
    (Set.preimage (β := ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
      (fun (x : ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type)) ↦
        match (motive := ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type) → ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
          x with
        | { down := ⟨p, hp⟩ } =>
          {
            down :=
              Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) p sorry })
      (ULift.down ⁻¹'
        Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
            Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
          r))
Tactic: tauto
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Sphere (goal)
Goal:
n : ℕ
t : Set (ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
h_1 : IsOpen t
s : Set (↑(Metric.closedBall 0 1) : Type)
r : Set (EuclideanSpace ℝ (Fin n))
hro : IsOpen r
hrs :
  Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
        Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
      r =
    s
hst : ULift.down ⁻¹' s = t
h :
  IsOpen (X := (↑(Metric.ball 0 1) : Type))
      (Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
          Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.ball 0 1) x) ⁻¹'
        r) ∧
    Eq.{u_1 + 1} (α := Set (ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type)))
      (ULift.down ⁻¹'
        Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
            Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.ball 0 1) x) ⁻¹'
          r)
      (Set.preimage (β := ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
        (fun (x : ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type)) ↦
          match (motive :=
            ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type) → ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type)) x with
          | { down := ⟨p, hp⟩ } =>
            {
              down :=
                Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                  Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) p ⋯ })
        (ULift.down ⁻¹'
          Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
              Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
            r))
⊢ ∃ (t : Set (↑(Metric.ball 0 1) : Type)),
    IsOpen t ∧
      ULift.down ⁻¹' t =
        Set.preimage (β := ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type))
          (fun (x : ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type)) ↦
            match (motive :=
              ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type) → ULift.{u_1, 0} (↑(Metric.closedBall 0 1) : Type)) x with
            | { down := ⟨p, hp⟩ } =>
              {
                down :=
                  Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                    Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) p ⋯ })
          (ULift.down ⁻¹'
            Subtype.val (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.closedBall 0 1) x) ⁻¹'
              r)
have w : Set (↑(Metric.ball 0 1) : Type)
Tactic: tauto
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Sphere (goal)
Goal:
n : ℕ
x : EuclideanSpace ℝ (Fin n)
hx : Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.sphere 0 1) x
⊢ ∀ ⦃a₂ : (↑(∂𝔻 n) : Type u_1)⦄,
    Eq (α := (↑(𝔻 n) : Type u_1))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
                (diskBoundaryInclusion n) :
              (a : (↑(∂𝔻 n) : Type u_1)) → (↑(𝔻 n) : Type u_1))
            {
              down :=
                Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                  Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.sphere 0 1) x) x hx :
              ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type) } :
          (↑(𝔻 n) : Type u_1))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
                (diskBoundaryInclusion n) :
              (a : (↑(∂𝔻 n) : Type u_1)) → (↑(𝔻 n) : Type u_1))
            a₂ :
          (↑(𝔻 n) : Type u_1)) →
      {
          down :=
            Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
              Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.sphere 0 1) x) x hx :
          ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type) } =
        a₂
have this : ∀ (y : EuclideanSpace ℝ (Fin n)) (hy : Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.sphere 0 1) y),
  Eq (α := (↑(𝔻 n) : Type u_1))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (diskBoundaryInclusion n) :
            (a : (↑(∂𝔻 n) : Type u_1)) → (↑(𝔻 n) : Type u_1))
          {
            down :=
              Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.sphere 0 1) x) x hx :
            ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type) } :
        (↑(𝔻 n) : Type u_1))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (diskBoundaryInclusion n) :
            (a : (↑(∂𝔻 n) : Type u_1)) → (↑(𝔻 n) : Type u_1))
          {
            down :=
              Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.sphere 0 1) x) y hy :
            ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type) } :
        (↑(𝔻 n) : Type u_1)) →
    {
        down :=
          Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
            Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.sphere 0 1) x) x hx :
        ULift.{u_1, 0} (↑(Metric.sphere 0 1) : Type) } =
      {
        down :=
          Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
            Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.sphere 0 1) x) y hy }
Tactic: ⟨y, hy⟩
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Sphere (goal)
Goal:
n : ℕ
x : EuclideanSpace ℝ (Fin n)
hx : Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.ball 0 1) x
⊢ ∀ ⦃a₂ : (↑(𝔹 n) : Type u_1)⦄,
    Eq (α := (↑(𝔻 n) : Type u_1))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
                (ballInclusion n) :
              (a : (↑(𝔹 n) : Type u_1)) → (↑(𝔻 n) : Type u_1))
            {
              down :=
                Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                  Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.ball 0 1) x) x hx :
              ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type) } :
          (↑(𝔻 n) : Type u_1))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
                (ballInclusion n) :
              (a : (↑(𝔹 n) : Type u_1)) → (↑(𝔻 n) : Type u_1))
            a₂ :
          (↑(𝔻 n) : Type u_1)) →
      {
          down :=
            Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
              Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.ball 0 1) x) x hx :
          ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type) } =
        a₂
have this : ∀ (y : EuclideanSpace ℝ (Fin n)) (hy : Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.ball 0 1) y),
  Eq (α := (↑(𝔻 n) : Type u_1))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (ballInclusion n) :
            (a : (↑(𝔹 n) : Type u_1)) → (↑(𝔻 n) : Type u_1))
          {
            down :=
              Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.ball 0 1) x) x hx :
            ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type) } :
        (↑(𝔻 n) : Type u_1))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (ballInclusion n) :
            (a : (↑(𝔹 n) : Type u_1)) → (↑(𝔻 n) : Type u_1))
          {
            down :=
              Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
                Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.ball 0 1) x) y hy :
            ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type) } :
        (↑(𝔻 n) : Type u_1)) →
    {
        down :=
          Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
            Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.ball 0 1) x) x hx :
        ULift.{u_1, 0} (↑(Metric.ball 0 1) : Type) } =
      {
        down :=
          Subtype.mk (p := fun (x : EuclideanSpace ℝ (Fin n)) ↦
            Membership.mem (γ := Set (EuclideanSpace ℝ (Fin n))) (Metric.ball 0 1) x) y hy }
Tactic: ⟨y, hy⟩
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Sphere (goal)
Goal:
n : ℕ
⊢ CompactSpace.{u_1} (↑(𝔻 n) : Type u_1)
have convert_1 : CompactSpace (↑(Metric.closedBall 0 1) : Type)
Tactic: convert! Homeomorph.compactSpace Homeomorph.ulift.symm
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Sphere (goal)
Goal:
n : ℕ
⊢ CompactSpace.{u_1} (↑(∂𝔻 n) : Type u_1)
have convert_1 : CompactSpace (↑(Metric.sphere 0 1) : Type)
Tactic: convert! Homeomorph.compactSpace Homeomorph.ulift.symm
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.ULift (goal)
Goal:
⊢ PreservesLimitsOfSize.{w', w, u, max u v, u + 1, max (u + 1) (v + 1)} uliftFunctor
have refine : ∀ {J : Type w} [inst : Category.{w', w} J] {K : CategoryTheory.Functor.{w', u, w, u + 1} J TopCat} {c : Cone K}
  (hc : IsLimit c), Nonempty (IsLimit (uliftFunctor.mapCone c))
Tactic: refine ⟨⟨fun {K} ↦ ⟨fun {c} hc ↦ ?_⟩⟩⟩
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.ULift (goal)
Goal:
J : Type w
inst : Category.{w', w} J
K : CategoryTheory.Functor.{w', u, w, u + 1} J TopCat
c : Cone K
hc : IsLimit c
⊢ Nonempty (IsLimit (uliftFunctor.mapCone c))
have rw : Eq (α := TopologicalSpace (↑(uliftFunctor.mapCone c).pt : Type (max u v))) (uliftFunctor.mapCone c).pt.str
  (⨅ (j : J),
    TopologicalSpace.induced (Y := (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
            carrier) ((uliftFunctor.mapCone c).π.app j)) :
        (a : (↑(((Functor.const J).obj (uliftFunctor.mapCone c).pt).obj j) : Type (max u v))) →
          (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
      ((K ⋙ uliftFunctor).obj j).str)
Tactic: rw [nonempty_isLimit_iff_eq_induced]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.ULift (goal)
Goal:
J : Type w
inst : Category.{w', w} J
K : CategoryTheory.Functor.{w', u, w, u + 1} J TopCat
c : Cone K
hc : IsLimit c
rw :
  Eq (α := TopologicalSpace (↑(uliftFunctor.mapCone c).pt : Type (max u v))) (uliftFunctor.mapCone c).pt.str
    (⨅ (j : J),
      TopologicalSpace.induced (Y := (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
              carrier) ((uliftFunctor.mapCone c).π.app j)) :
          (a : (↑(((Functor.const J).obj (uliftFunctor.mapCone c).pt).obj j) : Type (max u v))) →
            (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
        ((K ⋙ uliftFunctor).obj j).str)
⊢ Nonempty (IsLimit (uliftFunctor.mapCone c))
have hc₁ : IsLimit (F :=
  (K ⋙ uliftFunctor) ⋙
    forget TopCat (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier))
  (Functor.mapCone
    (forget TopCat (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier))
    (uliftFunctor.mapCone c))
Tactic: rw [nonempty_isLimit_iff_eq_induced]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.ULift (goal)
Goal:
J : Type w
inst : Category.{w', w} J
K : CategoryTheory.Functor.{w', u, w, u + 1} J TopCat
c : Cone K
hc : IsLimit c
⊢ Eq (α := TopologicalSpace (↑(uliftFunctor.mapCone c).pt : Type (max u v))) (uliftFunctor.mapCone c).pt.str
    (⨅ (j : J),
      TopologicalSpace.induced (Y := (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
              carrier) ((uliftFunctor.mapCone c).π.app j)) :
          (a : (↑(((Functor.const J).obj (uliftFunctor.mapCone c).pt).obj j) : Type (max u v))) →
            (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
        ((K ⋙ uliftFunctor).obj j).str)
have refine_1 : LE.le (α := TopologicalSpace (↑(uliftFunctor.mapCone c).pt : Type (max u v))) (uliftFunctor.mapCone c).pt.str
  (⨅ (j : J),
    TopologicalSpace.induced (Y := (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
            carrier) ((uliftFunctor.mapCone c).π.app j)) :
        (a : (↑(((Functor.const J).obj (uliftFunctor.mapCone c).pt).obj j) : Type (max u v))) →
          (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
      ((K ⋙ uliftFunctor).obj j).str)
Tactic: refine le_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.ULift (goal)
Goal:
J : Type w
inst : Category.{w', w} J
K : CategoryTheory.Functor.{w', u, w, u + 1} J TopCat
c : Cone K
hc : IsLimit c
refine_1 :
  LE.le (α := TopologicalSpace (↑(uliftFunctor.mapCone c).pt : Type (max u v))) (uliftFunctor.mapCone c).pt.str
    (⨅ (j : J),
      TopologicalSpace.induced (Y := (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
              carrier) ((uliftFunctor.mapCone c).π.app j)) :
          (a : (↑(((Functor.const J).obj (uliftFunctor.mapCone c).pt).obj j) : Type (max u v))) →
            (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
        ((K ⋙ uliftFunctor).obj j).str)
⊢ Eq (α := TopologicalSpace (↑(uliftFunctor.mapCone c).pt : Type (max u v))) (uliftFunctor.mapCone c).pt.str
    (⨅ (j : J),
      TopologicalSpace.induced (Y := (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
              carrier) ((uliftFunctor.mapCone c).π.app j)) :
          (a : (↑(((Functor.const J).obj (uliftFunctor.mapCone c).pt).obj j) : Type (max u v))) →
            (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
        ((K ⋙ uliftFunctor).obj j).str)
have refine_2 : LE.le (α := TopologicalSpace (↑(uliftFunctor.mapCone c).pt : Type (max u v)))
  (⨅ (j : J),
    TopologicalSpace.induced (Y := (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
            carrier) ((uliftFunctor.mapCone c).π.app j)) :
        (a : (↑(((Functor.const J).obj (uliftFunctor.mapCone c).pt).obj j) : Type (max u v))) →
          (↑((K ⋙ uliftFunctor).obj j) : Type (max u v)))
      ((K ⋙ uliftFunctor).obj j).str)
  (uliftFunctor.mapCone c).pt.str
Tactic: refine le_antisymm ?_ ?_