l3lab/ntp-mathlib huggingface dataset:
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
⊢ ∀ (x₀ : A) {U : Set A},
    U ∈
        { sets := {U | ∃ i, U = ↑(B i)}, nonempty := (_ : ∃ x, x ∈ {U | ∃ i, U = ↑(B i)}),
            inter_sets :=
              (_ :
                ∀ {x y : Set A},
                  x ∈ {U | ∃ i, U = ↑(B i)} →
                    y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y) }.sets →
      ∃
        V ∈
          { sets := {U | ∃ i, U = ↑(B i)}, nonempty := (_ : ∃ x, x ∈ {U | ∃ i, U = ↑(B i)}),
              inter_sets :=
                (_ :
                  ∀ {x y : Set A},
                    x ∈ {U | ∃ i, U = ↑(B i)} →
                      y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y) }.sets,
        V ⊆ (fun x => x₀ + x + -x₀) ⁻¹' U

Next tactic:
rintro x₀ _ ⟨i, rfl⟩
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    

Proof state:
case intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i : ι
⊢ ∃
    V ∈
      { sets := {U | ∃ i, U = ↑(B i)}, nonempty := (_ : ∃ x, x ∈ {U | ∃ i, U = ↑(B i)}),
          inter_sets :=
            (_ :
              ∀ {x y : Set A},
                x ∈ {U | ∃ i, U = ↑(B i)} → y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y) }.sets,
    V ⊆ (fun x => x₀ + x + -x₀) ⁻¹' ↑(B i)

Next tactic:
use B i
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    

Proof state:
case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i : ι
⊢ ↑(B i) ∈
      { sets := {U | ∃ i, U = ↑(B i)}, nonempty := (_ : ∃ x, x ∈ {U | ∃ i, U = ↑(B i)}),
          inter_sets :=
            (_ :
              ∀ {x y : Set A},
                x ∈ {U | ∃ i, U = ↑(B i)} → y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y) }.sets ∧
    ↑(B i) ⊆ (fun x => x₀ + x + -x₀) ⁻¹' ↑(B i)

Next tactic:
constructor
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · 

Proof state:
case h.left
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i : ι
⊢ ↑(B i) ∈
    { sets := {U | ∃ i, U = ↑(B i)}, nonempty := (_ : ∃ x, x ∈ {U | ∃ i, U = ↑(B i)}),
        inter_sets :=
          (_ :
            ∀ {x y : Set A},
              x ∈ {U | ∃ i, U = ↑(B i)} → y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y) }.sets

Next tactic:
use i
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · 

Proof state:
case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i : ι
⊢ ↑(B i) ⊆ (fun x => x₀ + x + -x₀) ⁻¹' ↑(B i)

Next tactic:
simp
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
⊢ ∀ {U : Set A}, U ∈ AddGroupFilterBasis.toFilterBasis.sets → ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V * V ⊆ U

Next tactic:
rintro _ ⟨i, rfl⟩
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    

Proof state:
case intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i : ι
⊢ ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V * V ⊆ ↑(B i)

Next tactic:
cases' hB.mul i with k hk
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    

Proof state:
case intro.intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i k : ι
hk : ↑(B k) * ↑(B k) ⊆ ↑(B i)
⊢ ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V * V ⊆ ↑(B i)

Next tactic:
use B k
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    

Proof state:
case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i k : ι
hk : ↑(B k) * ↑(B k) ⊆ ↑(B i)
⊢ ↑(B k) ∈ AddGroupFilterBasis.toFilterBasis.sets ∧ ↑(B k) * ↑(B k) ⊆ ↑(B i)

Next tactic:
constructor
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · 

Proof state:
case h.left
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i k : ι
hk : ↑(B k) * ↑(B k) ⊆ ↑(B i)
⊢ ↑(B k) ∈ AddGroupFilterBasis.toFilterBasis.sets

Next tactic:
use k
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · 

Proof state:
case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i k : ι
hk : ↑(B k) * ↑(B k) ⊆ ↑(B i)
⊢ ↑(B k) * ↑(B k) ⊆ ↑(B i)

Next tactic:
exact hk
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_left' := by
    

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
⊢ ∀ (x₀ : A) {U : Set A},
    U ∈ AddGroupFilterBasis.toFilterBasis.sets →
      ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (fun x => x₀ * x) ⁻¹' U

Next tactic:
rintro x₀ _ ⟨i, rfl⟩
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_left' := by
    rintro x₀ _ ⟨i, rfl⟩
    

Proof state:
case intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i : ι
⊢ ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (fun x => x₀ * x) ⁻¹' ↑(B i)

Next tactic:
cases' hB.leftMul x₀ i with k hk
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_left' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.leftMul x₀ i with k hk
    

Proof state:
case intro.intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i k : ι
hk : ↑(B k) ⊆ (fun x => x₀ * x) ⁻¹' ↑(B i)
⊢ ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (fun x => x₀ * x) ⁻¹' ↑(B i)

Next tactic:
use B k
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_left' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.leftMul x₀ i with k hk
    use B k
    

Proof state:
case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i k : ι
hk : ↑(B k) ⊆ (fun x => x₀ * x) ⁻¹' ↑(B i)
⊢ ↑(B k) ∈ AddGroupFilterBasis.toFilterBasis.sets ∧ ↑(B k) ⊆ (fun x => x₀ * x) ⁻¹' ↑(B i)

Next tactic:
constructor
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_left' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.leftMul x₀ i with k hk
    use B k
    constructor
    · 

Proof state:
case h.left
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i k : ι
hk : ↑(B k) ⊆ (fun x => x₀ * x) ⁻¹' ↑(B i)
⊢ ↑(B k) ∈ AddGroupFilterBasis.toFilterBasis.sets

Next tactic:
use k
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_left' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.leftMul x₀ i with k hk
    use B k
    constructor
    · use k
    · 

Proof state:
case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i k : ι
hk : ↑(B k) ⊆ (fun x => x₀ * x) ⁻¹' ↑(B i)
⊢ ↑(B k) ⊆ (fun x => x₀ * x) ⁻¹' ↑(B i)

Next tactic:
exact hk
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_left' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.leftMul x₀ i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_right' := by
    

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
⊢ ∀ (x₀ : A) {U : Set A},
    U ∈ AddGroupFilterBasis.toFilterBasis.sets →
      ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (fun x => x * x₀) ⁻¹' U

Next tactic:
rintro x₀ _ ⟨i, rfl⟩