awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Algebra.RestrictedProduct.Units (goal)
Goal:
ι : Type u_1
R : ι → Type u_2
inst : (i : ι) → Monoid (R i)
S : ι → Type u_3
inst_1 : (i : ι) → SetLike.{u_3, u_2} (S i) (R i)
inst_2 : ∀ (i : ι), SubmonoidClass.{u_3, u_2} (S i) (R i)
B : (i : ι) → S i
𝓕 : Filter ι
x : Πʳ (i : ι), [R i, (↑(B i) : Set ((fun (i : ι) ↦ R i) i))]_[𝓕]
b : Πʳ (i : ι), [R i, (↑(B i) : Set (R i))]_[𝓕]
hb :
  ∀ (x_1 : ι),
    (x * b : (a : ι) → R a) x_1 = (1 : (a : ι) → R a) x_1 ∧ (b * x : (a : ι) → R a) x_1 = (1 : (a : ι) → R a) x_1
⊢ ∀ᶠ (i : ι) in 𝓕,
    ∃ (h : (x : (a : ι) → R a) i ∈ B i) (b : ↥(B i)),
      Subtype.mk (p := fun (x : R i) ↦ x ∈ B i) ((x : (a : ι) → R a) i) ⋯ * b = 1 ∧
        b * Subtype.mk (p := fun (x : R i) ↦ x ∈ B i) ((x : (a : ι) → R a) i) ⋯ = 1
have h : ∀ (a : ι),
  Membership.mem (γ := Set ι)
    {x_1 : ι |
      Membership.mem (γ := Set ι)
          {x_2 : ι |
            (fun (i : ι) ↦
                Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                  ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i) ((↑x : (i : ι) → (fun (i : ι) ↦ R i) i) i))
              x_2}
          x_1 →
        Membership.mem (γ := Set ι)
          {x_2 : ι |
            Membership.mem (γ := Set ι)
                {x : ι |
                  (fun (i : ι) ↦
                      Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i)) ((fun (i : ι) ↦ (↑(B i) : Set (R i))) i)
                        ((↑b : (i : ι) → (fun (i : ι) ↦ R i) i) i))
                    x}
                x_2 →
              Membership.mem (γ := Set ι)
                {x_3 : ι |
                  (fun (i : ι) ↦
                      ∃ (h : (x : (a : ι) → R a) i ∈ B i) (b : ↥(B i)),
                        Subtype.mk (p := fun (x : R i) ↦ x ∈ B i) ((x : (a : ι) → R a) i) sorry * b = 1 ∧
                          b * Subtype.mk (p := fun (x : R i) ↦ x ∈ B i) ((x : (a : ι) → R a) i) sorry = 1)
                    x_3}
                x_2}
          x_1}
    a
Tactic: filter_upwards [x.2, b.2] using
    fun i hx hb ↦ ⟨hx, ⟨b i, hb⟩, by simp_all [← SetLike.coe_eq_coe]⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.RestrictedProduct.Units (goal)
Goal:
ι : Type u_1
R : ι → Type u_2
inst : (i : ι) → Monoid (R i)
S : ι → Type u_3
inst_1 : (i : ι) → SetLike.{u_3, u_2} (S i) (R i)
inst_2 : ∀ (i : ι), SubmonoidClass.{u_3, u_2} (S i) (R i)
B : (i : ι) → S i
𝓕 : Filter ι
x : Πʳ (i : ι), [R i, (↑(B i) : Set ((fun (i : ι) ↦ R i) i))]_[𝓕]
b : Πʳ (i : ι), [R i, (↑(B i) : Set (R i))]_[𝓕]
hb :
  ∀ (x_1 : ι),
    (x * b : (a : ι) → R a) x_1 = (1 : (a : ι) → R a) x_1 ∧ (b * x : (a : ι) → R a) x_1 = (1 : (a : ι) → R a) x_1
h :
  ∀ (a : ι),
    Membership.mem (γ := Set ι)
      {x_1 : ι |
        Membership.mem (γ := Set ι)
            {x_2 : ι |
              (fun (i : ι) ↦
                  Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                    ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
                    ((↑x : (i : ι) → (fun (i : ι) ↦ R i) i) i))
                x_2}
            x_1 →
          Membership.mem (γ := Set ι)
            {x_2 : ι |
              Membership.mem (γ := Set ι)
                  {x : ι |
                    (fun (i : ι) ↦
                        Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i)) ((fun (i : ι) ↦ (↑(B i) : Set (R i))) i)
                          ((↑b : (i : ι) → (fun (i : ι) ↦ R i) i) i))
                      x}
                  x_2 →
                Membership.mem (γ := Set ι)
                  {x_3 : ι |
                    (fun (i : ι) ↦
                        ∃ (h : (x : (a : ι) → R a) i ∈ B i) (b : ↥(B i)),
                          Subtype.mk (p := fun (x : R i) ↦ x ∈ B i) ((x : (a : ι) → R a) i) ⋯ * b = 1 ∧
                            b * Subtype.mk (p := fun (x : R i) ↦ x ∈ B i) ((x : (a : ι) → R a) i) ⋯ = 1)
                      x_3}
                  x_2}
            x_1}
      a
⊢ ∀ᶠ (i : ι) in 𝓕,
    ∃ (h : (x : (a : ι) → R a) i ∈ B i) (b : ↥(B i)),
      Subtype.mk (p := fun (x : R i) ↦ x ∈ B i) ((x : (a : ι) → R a) i) ⋯ * b = 1 ∧
        b * Subtype.mk (p := fun (x : R i) ↦ x ∈ B i) ((x : (a : ι) → R a) i) ⋯ = 1
have h₁ : ∀ (a : ι),
  (↑x : (i : ι) → R i) a ∈ (↑(B a) : Set (R a)) →
    (↑b : (i : ι) → R i) a ∈ (↑(B a) : Set (R a)) →
      ∃ (h : (x : (a : ι) → R a) a ∈ B a) (b : ↥(B a)),
        Subtype.mk (p := fun (x : R a) ↦ x ∈ B a) ((x : (a : ι) → R a) a) sorry * b = 1 ∧
          b * Subtype.mk (p := fun (x : R a) ↦ x ∈ B a) ((x : (a : ι) → R a) a) sorry = 1
Tactic: filter_upwards [x.2, b.2] using
    fun i hx hb ↦ ⟨hx, ⟨b i, hb⟩, by simp_all [← SetLike.coe_eq_coe]⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.RestrictedProduct.Units (goal)
Goal:
ι : Type u_1
R : ι → Type u_2
inst : (i : ι) → Monoid (R i)
S : ι → Type u_3
inst_1 : (i : ι) → SetLike.{u_3, u_2} (S i) (R i)
inst_2 : ∀ (i : ι), SubmonoidClass.{u_3, u_2} (S i) (R i)
B : (i : ι) → S i
𝓕 : Filter ι
x : (i : ι) → (R i)ˣ
hx : ∀ᶠ (i : ι) in 𝓕, Membership.mem (γ := Subgroup (R i)ˣ) (Submonoid.units (Submonoid.ofClass (B i))) (x i)
⊢ ∀ᶠ (i : ι) in 𝓕,
    Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i)) ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
      ((fun (i : ι) ↦ (↑(x i) : R i)) i)
have h : ∀ (a : ι),
  Membership.mem (γ := Set ι)
    {x_1 : ι |
      Membership.mem (γ := Set ι)
          {x_2 : ι |
            (fun (i : ι) ↦ Membership.mem (γ := Subgroup (R i)ˣ) (Submonoid.units (Submonoid.ofClass (B i))) (x i)) x_2}
          x_1 →
        Membership.mem (γ := Set ι)
          {x_2 : ι |
            (fun (i : ι) ↦
                Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                  ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i) ((fun (i : ι) ↦ (↑(x i) : R i)) i))
              x_2}
          x_1}
    a
Tactic: filter_upwards [hx] using fun i hi ↦ hi.1
-- Lean 4 proof step from Mathlib.Topology.Algebra.RestrictedProduct.Units (goal)
Goal:
ι : Type u_1
R : ι → Type u_2
inst : (i : ι) → Monoid (R i)
S : ι → Type u_3
inst_1 : (i : ι) → SetLike.{u_3, u_2} (S i) (R i)
inst_2 : ∀ (i : ι), SubmonoidClass.{u_3, u_2} (S i) (R i)
B : (i : ι) → S i
𝓕 : Filter ι
x : (i : ι) → (R i)ˣ
hx : ∀ᶠ (i : ι) in 𝓕, Membership.mem (γ := Subgroup (R i)ˣ) (Submonoid.units (Submonoid.ofClass (B i))) (x i)
h :
  ∀ (a : ι),
    Membership.mem (γ := Set ι)
      {x_1 : ι |
        Membership.mem (γ := Set ι)
            {x_2 : ι |
              (fun (i : ι) ↦ Membership.mem (γ := Subgroup (R i)ˣ) (Submonoid.units (Submonoid.ofClass (B i))) (x i))
                x_2}
            x_1 →
          Membership.mem (γ := Set ι)
            {x_2 : ι |
              (fun (i : ι) ↦
                  Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                    ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i) ((fun (i : ι) ↦ (↑(x i) : R i)) i))
                x_2}
            x_1}
      a
⊢ ∀ᶠ (i : ι) in 𝓕,
    Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i)) ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
      ((fun (i : ι) ↦ (↑(x i) : R i)) i)
have h₁ : ∀ (a : ι),
  Membership.mem (γ := Subgroup (R a)ˣ) (Submonoid.units (Submonoid.ofClass (B a))) (x a) →
    (↑(x a) : R a) ∈ (↑(B a) : Set (R a))
Tactic: filter_upwards [hx] using fun i hi ↦ hi.1
-- Lean 4 proof step from Mathlib.Topology.Algebra.RestrictedProduct.Units (goal)
Goal:
ι : Type u_1
R : ι → Type u_2
inst : (i : ι) → Monoid (R i)
S : ι → Type u_3
inst_1 : (i : ι) → SetLike.{u_3, u_2} (S i) (R i)
inst_2 : ∀ (i : ι), SubmonoidClass.{u_3, u_2} (S i) (R i)
B : (i : ι) → S i
𝓕 : Filter ι
x : (i : ι) → (R i)ˣ
hx : ∀ᶠ (i : ι) in 𝓕, Membership.mem (γ := Subgroup (R i)ˣ) (Submonoid.units (Submonoid.ofClass (B i))) (x i)
⊢ ∀ᶠ (i : ι) in 𝓕,
    Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i)) ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
      ((fun (i : ι) ↦ (↑(x i)⁻¹ : R i)) i)
have h : ∀ (a : ι),
  Membership.mem (γ := Set ι)
    {x_1 : ι |
      Membership.mem (γ := Set ι)
          {x_2 : ι |
            (fun (i : ι) ↦ Membership.mem (γ := Subgroup (R i)ˣ) (Submonoid.units (Submonoid.ofClass (B i))) (x i)) x_2}
          x_1 →
        Membership.mem (γ := Set ι)
          {x_2 : ι |
            (fun (i : ι) ↦
                Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                  ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i) ((fun (i : ι) ↦ (↑(x i)⁻¹ : R i)) i))
              x_2}
          x_1}
    a
Tactic: filter_upwards [hx] using fun i hi ↦ hi.2
-- Lean 4 proof step from Mathlib.Topology.Algebra.RestrictedProduct.Units (goal)
Goal:
ι : Type u_1
R : ι → Type u_2
inst : (i : ι) → Monoid (R i)
S : ι → Type u_3
inst_1 : (i : ι) → SetLike.{u_3, u_2} (S i) (R i)
inst_2 : ∀ (i : ι), SubmonoidClass.{u_3, u_2} (S i) (R i)
B : (i : ι) → S i
𝓕 : Filter ι
x : (i : ι) → (R i)ˣ
hx : ∀ᶠ (i : ι) in 𝓕, Membership.mem (γ := Subgroup (R i)ˣ) (Submonoid.units (Submonoid.ofClass (B i))) (x i)
h :
  ∀ (a : ι),
    Membership.mem (γ := Set ι)
      {x_1 : ι |
        Membership.mem (γ := Set ι)
            {x_2 : ι |
              (fun (i : ι) ↦ Membership.mem (γ := Subgroup (R i)ˣ) (Submonoid.units (Submonoid.ofClass (B i))) (x i))
                x_2}
            x_1 →
          Membership.mem (γ := Set ι)
            {x_2 : ι |
              (fun (i : ι) ↦
                  Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                    ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i) ((fun (i : ι) ↦ (↑(x i)⁻¹ : R i)) i))
                x_2}
            x_1}
      a
⊢ ∀ᶠ (i : ι) in 𝓕,
    Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i)) ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
      ((fun (i : ι) ↦ (↑(x i)⁻¹ : R i)) i)
have h₁ : ∀ (a : ι),
  Membership.mem (γ := Subgroup (R a)ˣ) (Submonoid.units (Submonoid.ofClass (B a))) (x a) →
    (↑(x a)⁻¹ : R a) ∈ (↑(B a) : Set (R a))
Tactic: filter_upwards [hx] using fun i hi ↦ hi.2
-- Lean 4 proof step from Mathlib.Topology.Algebra.RestrictedProduct.Units (goal)
Goal:
ι : Type u_1
R : ι → Type u_2
inst : (i : ι) → Monoid (R i)
S : ι → Type u_3
inst_1 : (i : ι) → SetLike.{u_3, u_2} (S i) (R i)
inst_2 : ∀ (i : ι), SubmonoidClass.{u_3, u_2} (S i) (R i)
B : (i : ι) → S i
𝓕 : Filter ι
x : (i : ι) → (R i)ˣ
hx : ∀ᶠ (i : ι) in 𝓕, Membership.mem (γ := Subgroup (R i)ˣ) (Submonoid.units (Submonoid.ofClass (B i))) (x i)
⊢ Subtype.mk (p := fun (x : (i : ι) → (fun (i : ι) ↦ R i) i) ↦
        ∀ᶠ (i : ι) in 𝓕,
          Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i)) ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
            (x i))
        (fun (i : ι) ↦ (↑(x i) : R i)) ⋯ *
      Subtype.mk (p := fun (x : (i : ι) → (fun (i : ι) ↦ R i) i) ↦
        ∀ᶠ (i : ι) in 𝓕,
          Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i)) ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
            (x i))
        (fun (i : ι) ↦ (↑(x i)⁻¹ : R i)) ⋯ =
    1
have h : ∀ (i : ι),
  (Subtype.mk (p := fun (x : (i : ι) → (fun (i : ι) ↦ R i) i) ↦
            ∀ᶠ (i : ι) in 𝓕,
              Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i) (x i))
            (fun (i : ι) ↦ (↑(x i) : R i)) sorry *
          Subtype.mk (p := fun (x : (i : ι) → (fun (i : ι) ↦ R i) i) ↦
            ∀ᶠ (i : ι) in 𝓕,
              Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i) (x i))
            (fun (i : ι) ↦ (↑(x i)⁻¹ : R i)) sorry :
        (a : ι) → R a)
      i =
    (1 : (a : ι) → R a) i
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Algebra.RestrictedProduct.Units (goal)
Goal:
ι : Type u_1
R : ι → Type u_2
inst : (i : ι) → Monoid (R i)
S : ι → Type u_3
inst_1 : (i : ι) → SetLike.{u_3, u_2} (S i) (R i)
inst_2 : ∀ (i : ι), SubmonoidClass.{u_3, u_2} (S i) (R i)
B : (i : ι) → S i
𝓕 : Filter ι
x : (i : ι) → (R i)ˣ
hx : ∀ᶠ (i : ι) in 𝓕, Membership.mem (γ := Subgroup (R i)ˣ) (Submonoid.units (Submonoid.ofClass (B i))) (x i)
⊢ Subtype.mk (p := fun (x : (i : ι) → (fun (i : ι) ↦ R i) i) ↦
        ∀ᶠ (i : ι) in 𝓕,
          Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i)) ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
            (x i))
        (fun (i : ι) ↦ (↑(x i)⁻¹ : R i)) ⋯ *
      Subtype.mk (p := fun (x : (i : ι) → (fun (i : ι) ↦ R i) i) ↦
        ∀ᶠ (i : ι) in 𝓕,
          Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i)) ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
            (x i))
        (fun (i : ι) ↦ (↑(x i) : R i)) ⋯ =
    1
have h : ∀ (i : ι),
  (Subtype.mk (p := fun (x : (i : ι) → (fun (i : ι) ↦ R i) i) ↦
            ∀ᶠ (i : ι) in 𝓕,
              Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i) (x i))
            (fun (i : ι) ↦ (↑(x i)⁻¹ : R i)) sorry *
          Subtype.mk (p := fun (x : (i : ι) → (fun (i : ι) ↦ R i) i) ↦
            ∀ᶠ (i : ι) in 𝓕,
              Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i) (x i))
            (fun (i : ι) ↦ (↑(x i) : R i)) sorry :
        (a : ι) → R a)
      i =
    (1 : (a : ι) → R a) i
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Algebra.RestrictedProduct.Units (goal)
Goal:
ι : Type u_1
R : ι → Type u_2
inst : (i : ι) → Monoid (R i)
S : ι → Type u_3
inst_1 : (i : ι) → SetLike.{u_3, u_2} (S i) (R i)
inst_2 : ∀ (i : ι), SubmonoidClass.{u_3, u_2} (S i) (R i)
B : (i : ι) → S i
𝓕 : Filter ι
x : Πʳ (i : ι), [R i, (↑(B i) : Set ((fun (i : ι) ↦ R i) i))]_[𝓕]ˣ
⊢ ∀ᶠ (i : ι) in 𝓕,
    Membership.mem.{u_2, u_2} (α := (fun (i : ι) ↦ Units.{u_2} (R i)) i) (γ :=
      Set ((fun (i : ι) ↦ Units.{u_2} (R i)) i))
      ((fun (i : ι) ↦ (↑(Submonoid.units (Submonoid.ofClass (B i))) : Set ((fun (i : ι) ↦ (R i)ˣ) i))) i)
      ((coeUnits (R := R) (S := S) :
            (a : Πʳ (i : ι), [R i, (↑(B i) : Set ((fun (i : ι) ↦ R i) i))]_[𝓕]ˣ) → (i : ι) → (R i)ˣ)
          x i :
        (R i)ˣ)
have h : ∀ (a : ι),
  Membership.mem (γ := Set ι)
    {x_1 : ι |
      Membership.mem (γ := Set ι)
          {x_2 : ι |
            (fun (i : ι) ↦
                Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                  ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
                  ((↑(↑x : Πʳ (i : ι), [R i, (↑(B i) : Set ((fun (i : ι) ↦ R i) i))]_[𝓕]) :
                      (i : ι) → (fun (i : ι) ↦ R i) i)
                    i))
              x_2}
          x_1 →
        Membership.mem (γ := Set ι)
          {x_2 : ι |
            Membership.mem (γ := Set ι)
                {x_3 : ι |
                  (fun (i : ι) ↦
                      Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                        ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
                        ((↑x.inv : (i : ι) → (fun (i : ι) ↦ R i) i) i))
                    x_3}
                x_2 →
              Membership.mem (γ := Set ι)
                {x_3 : ι |
                  (fun (i : ι) ↦
                      Membership.mem.{u_2, u_2} (α := (fun (i : ι) ↦ Units.{u_2} (R i)) i) (γ :=
                        Set ((fun (i : ι) ↦ Units.{u_2} (R i)) i))
                        ((fun (i : ι) ↦ (↑(Submonoid.units (Submonoid.ofClass (B i))) : Set ((fun (i : ι) ↦ (R i)ˣ) i)))
                          i)
                        ((coeUnits (R := R) (S := S) :
                              (a : Πʳ (i : ι), [R i, (↑(B i) : Set ((fun (i : ι) ↦ R i) i))]_[𝓕]ˣ) → (i : ι) → (R i)ˣ)
                            x i :
                          (R i)ˣ))
                    x_3}
                x_2}
          x_1}
    a
Tactic: filter_upwards [x.val.2, x.inv.2] using fun i hi hi' ↦ ⟨hi, hi'⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.RestrictedProduct.Units (goal)
Goal:
ι : Type u_1
R : ι → Type u_2
inst : (i : ι) → Monoid (R i)
S : ι → Type u_3
inst_1 : (i : ι) → SetLike.{u_3, u_2} (S i) (R i)
inst_2 : ∀ (i : ι), SubmonoidClass.{u_3, u_2} (S i) (R i)
B : (i : ι) → S i
𝓕 : Filter ι
x : Πʳ (i : ι), [R i, (↑(B i) : Set ((fun (i : ι) ↦ R i) i))]_[𝓕]ˣ
h :
  ∀ (a : ι),
    Membership.mem (γ := Set ι)
      {x_1 : ι |
        Membership.mem (γ := Set ι)
            {x_2 : ι |
              (fun (i : ι) ↦
                  Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                    ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
                    ((↑(↑x : Πʳ (i : ι), [R i, (↑(B i) : Set ((fun (i : ι) ↦ R i) i))]_[𝓕]) :
                        (i : ι) → (fun (i : ι) ↦ R i) i)
                      i))
                x_2}
            x_1 →
          Membership.mem (γ := Set ι)
            {x_2 : ι |
              Membership.mem (γ := Set ι)
                  {x_3 : ι |
                    (fun (i : ι) ↦
                        Membership.mem (γ := Set ((fun (i : ι) ↦ R i) i))
                          ((fun (i : ι) ↦ (↑(B i) : Set ((fun (i : ι) ↦ R i) i))) i)
                          ((↑x.inv : (i : ι) → (fun (i : ι) ↦ R i) i) i))
                      x_3}
                  x_2 →
                Membership.mem (γ := Set ι)
                  {x_3 : ι |
                    (fun (i : ι) ↦
                        Membership.mem.{u_2, u_2} (α := (fun (i : ι) ↦ Units.{u_2} (R i)) i) (γ :=
                          Set ((fun (i : ι) ↦ Units.{u_2} (R i)) i))
                          ((fun (i : ι) ↦
                              (↑(Submonoid.units (Submonoid.ofClass (B i))) : Set ((fun (i : ι) ↦ (R i)ˣ) i)))
                            i)
                          ((coeUnits (R := R) (S := S) :
                                (a : Πʳ (i : ι), [R i, (↑(B i) : Set ((fun (i : ι) ↦ R i) i))]_[𝓕]ˣ) → (i : ι) → (R i)ˣ)
                              x i :
                            (R i)ˣ))
                      x_3}
                  x_2}
            x_1}
      a
⊢ ∀ᶠ (i : ι) in 𝓕,
    Membership.mem.{u_2, u_2} (α := (fun (i : ι) ↦ Units.{u_2} (R i)) i) (γ :=
      Set ((fun (i : ι) ↦ Units.{u_2} (R i)) i))
      ((fun (i : ι) ↦ (↑(Submonoid.units (Submonoid.ofClass (B i))) : Set ((fun (i : ι) ↦ (R i)ˣ) i))) i)
      ((coeUnits (R := R) (S := S) :
            (a : Πʳ (i : ι), [R i, (↑(B i) : Set ((fun (i : ι) ↦ R i) i))]_[𝓕]ˣ) → (i : ι) → (R i)ˣ)
          x i :
        (R i)ˣ)
have h₁ : ∀ (a : ι),
  (↑(↑x : Πʳ (i : ι), [R i, (↑(B i) : Set (R i))]_[𝓕]) : (i : ι) → R i) a ∈ (↑(B a) : Set (R a)) →
    (↑x.inv : (i : ι) → R i) a ∈ (↑(B a) : Set (R a)) →
      Membership.mem (α := (R a)ˣ) (↑(Submonoid.units (Submonoid.ofClass (B a))) : Set (R a)ˣ)
        ((coeUnits (R := R) (S := S) : (a : Πʳ (i : ι), [R i, (↑(B i) : Set (R i))]_[𝓕]ˣ) → (i : ι) → (R i)ˣ) x a :
          (R a)ˣ)
Tactic: filter_upwards [x.val.2, x.inv.2] using fun i hi hi' ↦ ⟨hi, hi'⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Basic (goal)
Goal:
R : Type u_2
inst : NonUnitalNonAssocRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalAddGroup R
hmul :
  Tendsto (β := R) (uncurry fun (x1 x2 : R) ↦ x1 * x2) (SProd.sprod (α := Filter R) (β := Filter R) (𝓝 0) (𝓝 0)) (𝓝 0)
hmul_left : ∀ (x₀ : R), Tendsto (β := R) (fun (x : R) ↦ x₀ * x) (𝓝 0) (𝓝 0)
hmul_right : ∀ (x₀ : R), Tendsto (β := R) (fun (x : R) ↦ x * x₀) (𝓝 0) (𝓝 0)
⊢ Continuous[instTopologicalSpaceProd, inst_1] fun (p : R × R) ↦ p.1 * p.2
have refine_1 : ContinuousAt (Y := R)
  (fun (x : R × R) ↦ ((((AddMonoidHom.mul : R →+ R →+ R) : (a : R) → R →+ R) x.1 : R →+ R) : (a : R) → R) x.2) (0, 0)
Tactic: refine continuous_of_continuousAt_zero₂ (AddMonoidHom.mul (R := R)) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Basic (goal)
Goal:
R : Type u_2
inst : NonUnitalNonAssocRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalAddGroup R
hmul :
  Tendsto (β := R) (uncurry fun (x1 x2 : R) ↦ x1 * x2) (SProd.sprod (α := Filter R) (β := Filter R) (𝓝 0) (𝓝 0)) (𝓝 0)
hmul_left : ∀ (x₀ : R), Tendsto (β := R) (fun (x : R) ↦ x₀ * x) (𝓝 0) (𝓝 0)
hmul_right : ∀ (x₀ : R), Tendsto (β := R) (fun (x : R) ↦ x * x₀) (𝓝 0) (𝓝 0)
refine_1 :
  ContinuousAt (Y := R)
    (fun (x : R × R) ↦ ((((AddMonoidHom.mul : R →+ R →+ R) : (a : R) → R →+ R) x.1 : R →+ R) : (a : R) → R) x.2) (0, 0)
⊢ Continuous[instTopologicalSpaceProd, inst_1] fun (p : R × R) ↦ p.1 * p.2
have refine_2 : ∀ (x : R),
  ContinuousAt (X := R) (Y := R) (⇑(((AddMonoidHom.mul : R →+ R →+ R) : (a : R) → R →+ R) x : R →+ R) : (a : R) → R) 0
Tactic: refine continuous_of_continuousAt_zero₂ (AddMonoidHom.mul (R := R)) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Basic (goal)
Goal:
R : Type u_2
inst : NonUnitalNonAssocRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalAddGroup R
hmul :
  Tendsto (β := R) (uncurry fun (x1 x2 : R) ↦ x1 * x2) (SProd.sprod (α := Filter R) (β := Filter R) (𝓝 0) (𝓝 0)) (𝓝 0)
hmul_left : ∀ (x₀ : R), Tendsto (β := R) (fun (x : R) ↦ x₀ * x) (𝓝 0) (𝓝 0)
hmul_right : ∀ (x₀ : R), Tendsto (β := R) (fun (x : R) ↦ x * x₀) (𝓝 0) (𝓝 0)
refine_1 :
  ContinuousAt (Y := R)
    (fun (x : R × R) ↦ ((((AddMonoidHom.mul : R →+ R →+ R) : (a : R) → R →+ R) x.1 : R →+ R) : (a : R) → R) x.2) (0, 0)
refine_2 :
  ∀ (x : R),
    ContinuousAt (X := R) (Y := R) (⇑(((AddMonoidHom.mul : R →+ R →+ R) : (a : R) → R →+ R) x : R →+ R) : (a : R) → R) 0
⊢ Continuous[instTopologicalSpaceProd, inst_1] fun (p : R × R) ↦ p.1 * p.2
have refine_3 : ∀ (y : R),
  ContinuousAt (Y := R)
    (fun (x : R) ↦ ((((AddMonoidHom.mul : R →+ R →+ R) : (a : R) → R →+ R) x : R →+ R) : (a : R) → R) y) 0
Tactic: refine continuous_of_continuousAt_zero₂ (AddMonoidHom.mul (R := R)) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Basic (goal)
Goal:
R : Type u_1
inst : Ring R
f g : RingTopology R
h : f.toTopologicalSpace = g.toTopologicalSpace
⊢ f = g
have mk : ∀ (toTopologicalSpace : TopologicalSpace R) (toIsTopologicalRing : IsTopologicalRing R),
  { toTopologicalSpace := toTopologicalSpace, toIsTopologicalRing := toIsTopologicalRing :
          RingTopology R }.toTopologicalSpace =
      g.toTopologicalSpace →
    { toTopologicalSpace := toTopologicalSpace, toIsTopologicalRing := toIsTopologicalRing : RingTopology R } = g
Tactic: cases f
Cleared hypotheses: f, h
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Basic (goal)
Goal:
R : Type u_1
inst : Ring R
g : RingTopology R
toTopologicalSpace : TopologicalSpace R
toIsTopologicalRing : IsTopologicalRing R
h :
  { toTopologicalSpace := toTopologicalSpace, toIsTopologicalRing := toIsTopologicalRing :
        RingTopology R }.toTopologicalSpace =
    g.toTopologicalSpace
⊢ { toTopologicalSpace := toTopologicalSpace, toIsTopologicalRing := toIsTopologicalRing : RingTopology R } = g
have mk : ∀ (toTopologicalSpace_1 : TopologicalSpace R) (toIsTopologicalRing_1 : IsTopologicalRing R),
  { toTopologicalSpace := toTopologicalSpace, toIsTopologicalRing := toIsTopologicalRing :
          RingTopology R }.toTopologicalSpace =
      { toTopologicalSpace := toTopologicalSpace_1, toIsTopologicalRing := toIsTopologicalRing_1 :
          RingTopology R }.toTopologicalSpace →
    { toTopologicalSpace := toTopologicalSpace, toIsTopologicalRing := toIsTopologicalRing : RingTopology R } =
      { toTopologicalSpace := toTopologicalSpace_1, toIsTopologicalRing := toIsTopologicalRing_1 }
Tactic: cases g
Cleared hypotheses: g, h
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsArtinianRing R
⊢ Finite R
have obtain : ∀ (n : ℕ) (hn : HPow.hPow (α := Ideal R) (Ideal.jacobson ⊥) n = 0), Finite R
Tactic: obtain ⟨n, hn⟩ := IsArtinianRing.isNilpotent_jacobson_bot (R := R)
-- Lean 4 proof step from Mathlib.Topology.Algebra.Ring.Compact (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
inst_3 : CompactSpace R
inst_4 : T2Space R
inst_5 : IsArtinianRing R
n : ℕ
hn : HPow.hPow (α := Ideal R) (Ideal.jacobson ⊥) n = 0
⊢ HPow.hPow (α := Ideal R) (∏ p : PrimeSpectrum R, p.asIdeal) n = ⊥
have rw : HPow.hPow (α := Ideal R) (∏ p : PrimeSpectrum R, p.asIdeal) n ≤ HPow.hPow (α := Ideal R) (Ideal.jacobson ⊥) n
Tactic: rw [← le_bot_iff, ← Ideal.zero_eq_bot, ← hn]