l3lab/ntp-mathlib huggingface dataset:
/-- A general version of **Dirichlet's approximation theorem**.

See also `AddCircle.exists_norm_nsmul_le`. -/
lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
    [NormedAddCommGroup A] [CompactSpace A] [ConnectedSpace A]
    [MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
    (ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ/2))) :
    ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ δ := by
  have : IsFiniteMeasure μ := CompactSpace.isFiniteMeasure
  let B : Icc 0 n → Set A := fun j ↦ closedBall ((j : ℕ) • ξ) (δ/2)
  have hB : ∀ j, IsClosed (B j) := fun j ↦ isClosed_ball
  suffices : ¬ Pairwise (Disjoint on B)
  · obtain ⟨i, j, hij, x, hx⟩ := exists_lt_mem_inter_of_not_pairwise_disjoint this
    refine' ⟨j - i, ⟨le_tsub_of_add_le_left hij, _⟩, _⟩
    · simpa only [tsub_le_iff_right] using j.property.2.trans le_self_add
    · rw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), ← sub_eq_add_neg, ← dist_eq_norm]
      refine' (dist_triangle (↑j • ξ) x (↑i • ξ)).trans _
      linarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]
  by_contra h
  

Proof state:
case this
T : ℝ
hT : Fact (0 < T)
A : Type u_1
inst✝⁵ : NormedAddCommGroup A
inst✝⁴ : CompactSpace A
inst✝³ : ConnectedSpace A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure A
inst✝ : Measure.IsAddHaarMeasure μ
ξ : A
n : ℕ
hn : 0 < n
δ : ℝ
hδ : ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))
this : IsFiniteMeasure μ
B : ↑(Icc 0 n) → Set A := fun j => closedBall (↑j • ξ) (δ / 2)
hB : ∀ (j : ↑(Icc 0 n)), IsClosed (B j)
h : Pairwise (Disjoint on B)
⊢ False

Next tactic:
apply hn.ne'
/-- A general version of **Dirichlet's approximation theorem**.

See also `AddCircle.exists_norm_nsmul_le`. -/
lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
    [NormedAddCommGroup A] [CompactSpace A] [ConnectedSpace A]
    [MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
    (ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ/2))) :
    ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ δ := by
  have : IsFiniteMeasure μ := CompactSpace.isFiniteMeasure
  let B : Icc 0 n → Set A := fun j ↦ closedBall ((j : ℕ) • ξ) (δ/2)
  have hB : ∀ j, IsClosed (B j) := fun j ↦ isClosed_ball
  suffices : ¬ Pairwise (Disjoint on B)
  · obtain ⟨i, j, hij, x, hx⟩ := exists_lt_mem_inter_of_not_pairwise_disjoint this
    refine' ⟨j - i, ⟨le_tsub_of_add_le_left hij, _⟩, _⟩
    · simpa only [tsub_le_iff_right] using j.property.2.trans le_self_add
    · rw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), ← sub_eq_add_neg, ← dist_eq_norm]
      refine' (dist_triangle (↑j • ξ) x (↑i • ξ)).trans _
      linarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]
  by_contra h
  apply hn.ne'
  

Proof state:
case this
T : ℝ
hT : Fact (0 < T)
A : Type u_1
inst✝⁵ : NormedAddCommGroup A
inst✝⁴ : CompactSpace A
inst✝³ : ConnectedSpace A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure A
inst✝ : Measure.IsAddHaarMeasure μ
ξ : A
n : ℕ
hn : 0 < n
δ : ℝ
hδ : ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))
this : IsFiniteMeasure μ
B : ↑(Icc 0 n) → Set A := fun j => closedBall (↑j • ξ) (δ / 2)
hB : ∀ (j : ↑(Icc 0 n)), IsClosed (B j)
h : Pairwise (Disjoint on B)
⊢ n = 0

Next tactic:
have h' : ⋃ j, B j = univ := by
    rw [← (isClosed_iUnion_of_finite hB).measure_eq_univ_iff_eq (μ := μ)]
    refine' le_antisymm (μ.mono (subset_univ _)) _
    simp_rw [measure_iUnion h (fun _ ↦ measurableSet_closedBall), tsum_fintype,
      μ.addHaar_closedBall_center, Finset.sum_const, Finset.card_univ, Nat.card_fintypeIcc,
      tsub_zero]
    exact hδ
/-- A general version of **Dirichlet's approximation theorem**.

See also `AddCircle.exists_norm_nsmul_le`. -/
lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
    [NormedAddCommGroup A] [CompactSpace A] [ConnectedSpace A]
    [MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
    (ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ/2))) :
    ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ δ := by
  have : IsFiniteMeasure μ := CompactSpace.isFiniteMeasure
  let B : Icc 0 n → Set A := fun j ↦ closedBall ((j : ℕ) • ξ) (δ/2)
  have hB : ∀ j, IsClosed (B j) := fun j ↦ isClosed_ball
  suffices : ¬ Pairwise (Disjoint on B)
  · obtain ⟨i, j, hij, x, hx⟩ := exists_lt_mem_inter_of_not_pairwise_disjoint this
    refine' ⟨j - i, ⟨le_tsub_of_add_le_left hij, _⟩, _⟩
    · simpa only [tsub_le_iff_right] using j.property.2.trans le_self_add
    · rw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), ← sub_eq_add_neg, ← dist_eq_norm]
      refine' (dist_triangle (↑j • ξ) x (↑i • ξ)).trans _
      linarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]
  by_contra h
  apply hn.ne'
  have h' : ⋃ j, B j = univ := by
    

Proof state:
T : ℝ
hT : Fact (0 < T)
A : Type u_1
inst✝⁵ : NormedAddCommGroup A
inst✝⁴ : CompactSpace A
inst✝³ : ConnectedSpace A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure A
inst✝ : Measure.IsAddHaarMeasure μ
ξ : A
n : ℕ
hn : 0 < n
δ : ℝ
hδ : ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))
this : IsFiniteMeasure μ
B : ↑(Icc 0 n) → Set A := fun j => closedBall (↑j • ξ) (δ / 2)
hB : ∀ (j : ↑(Icc 0 n)), IsClosed (B j)
h : Pairwise (Disjoint on B)
⊢ ⋃ j, B j = univ

Next tactic:
rw [← (isClosed_iUnion_of_finite hB).measure_eq_univ_iff_eq (μ := μ)]
/-- A general version of **Dirichlet's approximation theorem**.

See also `AddCircle.exists_norm_nsmul_le`. -/
lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
    [NormedAddCommGroup A] [CompactSpace A] [ConnectedSpace A]
    [MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
    (ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ/2))) :
    ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ δ := by
  have : IsFiniteMeasure μ := CompactSpace.isFiniteMeasure
  let B : Icc 0 n → Set A := fun j ↦ closedBall ((j : ℕ) • ξ) (δ/2)
  have hB : ∀ j, IsClosed (B j) := fun j ↦ isClosed_ball
  suffices : ¬ Pairwise (Disjoint on B)
  · obtain ⟨i, j, hij, x, hx⟩ := exists_lt_mem_inter_of_not_pairwise_disjoint this
    refine' ⟨j - i, ⟨le_tsub_of_add_le_left hij, _⟩, _⟩
    · simpa only [tsub_le_iff_right] using j.property.2.trans le_self_add
    · rw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), ← sub_eq_add_neg, ← dist_eq_norm]
      refine' (dist_triangle (↑j • ξ) x (↑i • ξ)).trans _
      linarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]
  by_contra h
  apply hn.ne'
  have h' : ⋃ j, B j = univ := by
    rw [← (isClosed_iUnion_of_finite hB).measure_eq_univ_iff_eq (μ := μ)]
    

Proof state:
T : ℝ
hT : Fact (0 < T)
A : Type u_1
inst✝⁵ : NormedAddCommGroup A
inst✝⁴ : CompactSpace A
inst✝³ : ConnectedSpace A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure A
inst✝ : Measure.IsAddHaarMeasure μ
ξ : A
n : ℕ
hn : 0 < n
δ : ℝ
hδ : ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))
this : IsFiniteMeasure μ
B : ↑(Icc 0 n) → Set A := fun j => closedBall (↑j • ξ) (δ / 2)
hB : ∀ (j : ↑(Icc 0 n)), IsClosed (B j)
h : Pairwise (Disjoint on B)
⊢ ↑↑μ (⋃ i, B i) = ↑↑μ univ

Next tactic:
refine' le_antisymm (μ.mono (subset_univ _)) _
/-- A general version of **Dirichlet's approximation theorem**.

See also `AddCircle.exists_norm_nsmul_le`. -/
lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
    [NormedAddCommGroup A] [CompactSpace A] [ConnectedSpace A]
    [MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
    (ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ/2))) :
    ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ δ := by
  have : IsFiniteMeasure μ := CompactSpace.isFiniteMeasure
  let B : Icc 0 n → Set A := fun j ↦ closedBall ((j : ℕ) • ξ) (δ/2)
  have hB : ∀ j, IsClosed (B j) := fun j ↦ isClosed_ball
  suffices : ¬ Pairwise (Disjoint on B)
  · obtain ⟨i, j, hij, x, hx⟩ := exists_lt_mem_inter_of_not_pairwise_disjoint this
    refine' ⟨j - i, ⟨le_tsub_of_add_le_left hij, _⟩, _⟩
    · simpa only [tsub_le_iff_right] using j.property.2.trans le_self_add
    · rw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), ← sub_eq_add_neg, ← dist_eq_norm]
      refine' (dist_triangle (↑j • ξ) x (↑i • ξ)).trans _
      linarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]
  by_contra h
  apply hn.ne'
  have h' : ⋃ j, B j = univ := by
    rw [← (isClosed_iUnion_of_finite hB).measure_eq_univ_iff_eq (μ := μ)]
    refine' le_antisymm (μ.mono (subset_univ _)) _
    

Proof state:
T : ℝ
hT : Fact (0 < T)
A : Type u_1
inst✝⁵ : NormedAddCommGroup A
inst✝⁴ : CompactSpace A
inst✝³ : ConnectedSpace A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure A
inst✝ : Measure.IsAddHaarMeasure μ
ξ : A
n : ℕ
hn : 0 < n
δ : ℝ
hδ : ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))
this : IsFiniteMeasure μ
B : ↑(Icc 0 n) → Set A := fun j => closedBall (↑j • ξ) (δ / 2)
hB : ∀ (j : ↑(Icc 0 n)), IsClosed (B j)
h : Pairwise (Disjoint on B)
⊢ ↑↑μ univ ≤ ↑↑μ (⋃ i, B i)

Next tactic:
simp_rw [measure_iUnion h (fun _ ↦ measurableSet_closedBall), tsum_fintype,
      μ.addHaar_closedBall_center, Finset.sum_const, Finset.card_univ, Nat.card_fintypeIcc,
      tsub_zero]
/-- A general version of **Dirichlet's approximation theorem**.

See also `AddCircle.exists_norm_nsmul_le`. -/
lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
    [NormedAddCommGroup A] [CompactSpace A] [ConnectedSpace A]
    [MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
    (ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ/2))) :
    ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ δ := by
  have : IsFiniteMeasure μ := CompactSpace.isFiniteMeasure
  let B : Icc 0 n → Set A := fun j ↦ closedBall ((j : ℕ) • ξ) (δ/2)
  have hB : ∀ j, IsClosed (B j) := fun j ↦ isClosed_ball
  suffices : ¬ Pairwise (Disjoint on B)
  · obtain ⟨i, j, hij, x, hx⟩ := exists_lt_mem_inter_of_not_pairwise_disjoint this
    refine' ⟨j - i, ⟨le_tsub_of_add_le_left hij, _⟩, _⟩
    · simpa only [tsub_le_iff_right] using j.property.2.trans le_self_add
    · rw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), ← sub_eq_add_neg, ← dist_eq_norm]
      refine' (dist_triangle (↑j • ξ) x (↑i • ξ)).trans _
      linarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]
  by_contra h
  apply hn.ne'
  have h' : ⋃ j, B j = univ := by
    rw [← (isClosed_iUnion_of_finite hB).measure_eq_univ_iff_eq (μ := μ)]
    refine' le_antisymm (μ.mono (subset_univ _)) _
    simp_rw [measure_iUnion h (fun _ ↦ measurableSet_closedBall), tsum_fintype,
      μ.addHaar_closedBall_center, Finset.sum_const, Finset.card_univ, Nat.card_fintypeIcc,
      tsub_zero]
    

Proof state:
T : ℝ
hT : Fact (0 < T)
A : Type u_1
inst✝⁵ : NormedAddCommGroup A
inst✝⁴ : CompactSpace A
inst✝³ : ConnectedSpace A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure A
inst✝ : Measure.IsAddHaarMeasure μ
ξ : A
n : ℕ
hn : 0 < n
δ : ℝ
hδ : ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))
this : IsFiniteMeasure μ
B : ↑(Icc 0 n) → Set A := fun j => closedBall (↑j • ξ) (δ / 2)
hB : ∀ (j : ↑(Icc 0 n)), IsClosed (B j)
h : Pairwise (Disjoint on B)
⊢ ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))

Next tactic:
exact hδ
/-- A general version of **Dirichlet's approximation theorem**.

See also `AddCircle.exists_norm_nsmul_le`. -/
lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
    [NormedAddCommGroup A] [CompactSpace A] [ConnectedSpace A]
    [MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
    (ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ/2))) :
    ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ δ := by
  have : IsFiniteMeasure μ := CompactSpace.isFiniteMeasure
  let B : Icc 0 n → Set A := fun j ↦ closedBall ((j : ℕ) • ξ) (δ/2)
  have hB : ∀ j, IsClosed (B j) := fun j ↦ isClosed_ball
  suffices : ¬ Pairwise (Disjoint on B)
  · obtain ⟨i, j, hij, x, hx⟩ := exists_lt_mem_inter_of_not_pairwise_disjoint this
    refine' ⟨j - i, ⟨le_tsub_of_add_le_left hij, _⟩, _⟩
    · simpa only [tsub_le_iff_right] using j.property.2.trans le_self_add
    · rw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), ← sub_eq_add_neg, ← dist_eq_norm]
      refine' (dist_triangle (↑j • ξ) x (↑i • ξ)).trans _
      linarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]
  by_contra h
  apply hn.ne'
  have h' : ⋃ j, B j = univ := by
    rw [← (isClosed_iUnion_of_finite hB).measure_eq_univ_iff_eq (μ := μ)]
    refine' le_antisymm (μ.mono (subset_univ _)) _
    simp_rw [measure_iUnion h (fun _ ↦ measurableSet_closedBall), tsum_fintype,
      μ.addHaar_closedBall_center, Finset.sum_const, Finset.card_univ, Nat.card_fintypeIcc,
      tsub_zero]
    exact hδ
  

Proof state:
case this
T : ℝ
hT : Fact (0 < T)
A : Type u_1
inst✝⁵ : NormedAddCommGroup A
inst✝⁴ : CompactSpace A
inst✝³ : ConnectedSpace A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure A
inst✝ : Measure.IsAddHaarMeasure μ
ξ : A
n : ℕ
hn : 0 < n
δ : ℝ
hδ : ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))
this : IsFiniteMeasure μ
B : ↑(Icc 0 n) → Set A := fun j => closedBall (↑j • ξ) (δ / 2)
hB : ∀ (j : ↑(Icc 0 n)), IsClosed (B j)
h : Pairwise (Disjoint on B)
h' : ⋃ j, B j = univ
⊢ n = 0

Next tactic:
replace hδ : 0 ≤ δ/2 := by
    by_contra contra
    suffices : μ (closedBall 0 (δ/2)) = 0
    · apply isOpen_univ.measure_ne_zero μ univ_nonempty $ le_zero_iff.mp $ le_trans hδ _
      simp [this]
    rw [not_le, ← closedBall_eq_empty (x := (0 : A))] at contra
    simp [contra]
/-- A general version of **Dirichlet's approximation theorem**.

See also `AddCircle.exists_norm_nsmul_le`. -/
lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
    [NormedAddCommGroup A] [CompactSpace A] [ConnectedSpace A]
    [MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
    (ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ/2))) :
    ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ δ := by
  have : IsFiniteMeasure μ := CompactSpace.isFiniteMeasure
  let B : Icc 0 n → Set A := fun j ↦ closedBall ((j : ℕ) • ξ) (δ/2)
  have hB : ∀ j, IsClosed (B j) := fun j ↦ isClosed_ball
  suffices : ¬ Pairwise (Disjoint on B)
  · obtain ⟨i, j, hij, x, hx⟩ := exists_lt_mem_inter_of_not_pairwise_disjoint this
    refine' ⟨j - i, ⟨le_tsub_of_add_le_left hij, _⟩, _⟩
    · simpa only [tsub_le_iff_right] using j.property.2.trans le_self_add
    · rw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), ← sub_eq_add_neg, ← dist_eq_norm]
      refine' (dist_triangle (↑j • ξ) x (↑i • ξ)).trans _
      linarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]
  by_contra h
  apply hn.ne'
  have h' : ⋃ j, B j = univ := by
    rw [← (isClosed_iUnion_of_finite hB).measure_eq_univ_iff_eq (μ := μ)]
    refine' le_antisymm (μ.mono (subset_univ _)) _
    simp_rw [measure_iUnion h (fun _ ↦ measurableSet_closedBall), tsum_fintype,
      μ.addHaar_closedBall_center, Finset.sum_const, Finset.card_univ, Nat.card_fintypeIcc,
      tsub_zero]
    exact hδ
  replace hδ : 0 ≤ δ/2 := by
    

Proof state:
T : ℝ
hT : Fact (0 < T)
A : Type u_1
inst✝⁵ : NormedAddCommGroup A
inst✝⁴ : CompactSpace A
inst✝³ : ConnectedSpace A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure A
inst✝ : Measure.IsAddHaarMeasure μ
ξ : A
n : ℕ
hn : 0 < n
δ : ℝ
hδ : ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))
this : IsFiniteMeasure μ
B : ↑(Icc 0 n) → Set A := fun j => closedBall (↑j • ξ) (δ / 2)
hB : ∀ (j : ↑(Icc 0 n)), IsClosed (B j)
h : Pairwise (Disjoint on B)
h' : ⋃ j, B j = univ
⊢ 0 ≤ δ / 2

Next tactic:
by_contra contra
/-- A general version of **Dirichlet's approximation theorem**.

See also `AddCircle.exists_norm_nsmul_le`. -/
lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
    [NormedAddCommGroup A] [CompactSpace A] [ConnectedSpace A]
    [MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
    (ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ/2))) :
    ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ δ := by
  have : IsFiniteMeasure μ := CompactSpace.isFiniteMeasure
  let B : Icc 0 n → Set A := fun j ↦ closedBall ((j : ℕ) • ξ) (δ/2)
  have hB : ∀ j, IsClosed (B j) := fun j ↦ isClosed_ball
  suffices : ¬ Pairwise (Disjoint on B)
  · obtain ⟨i, j, hij, x, hx⟩ := exists_lt_mem_inter_of_not_pairwise_disjoint this
    refine' ⟨j - i, ⟨le_tsub_of_add_le_left hij, _⟩, _⟩
    · simpa only [tsub_le_iff_right] using j.property.2.trans le_self_add
    · rw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), ← sub_eq_add_neg, ← dist_eq_norm]
      refine' (dist_triangle (↑j • ξ) x (↑i • ξ)).trans _
      linarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]
  by_contra h
  apply hn.ne'
  have h' : ⋃ j, B j = univ := by
    rw [← (isClosed_iUnion_of_finite hB).measure_eq_univ_iff_eq (μ := μ)]
    refine' le_antisymm (μ.mono (subset_univ _)) _
    simp_rw [measure_iUnion h (fun _ ↦ measurableSet_closedBall), tsum_fintype,
      μ.addHaar_closedBall_center, Finset.sum_const, Finset.card_univ, Nat.card_fintypeIcc,
      tsub_zero]
    exact hδ
  replace hδ : 0 ≤ δ/2 := by
    by_contra contra
    

Proof state:
T : ℝ
hT : Fact (0 < T)
A : Type u_1
inst✝⁵ : NormedAddCommGroup A
inst✝⁴ : CompactSpace A
inst✝³ : ConnectedSpace A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure A
inst✝ : Measure.IsAddHaarMeasure μ
ξ : A
n : ℕ
hn : 0 < n
δ : ℝ
hδ : ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))
this : IsFiniteMeasure μ
B : ↑(Icc 0 n) → Set A := fun j => closedBall (↑j • ξ) (δ / 2)
hB : ∀ (j : ↑(Icc 0 n)), IsClosed (B j)
h : Pairwise (Disjoint on B)
h' : ⋃ j, B j = univ
contra : ¬0 ≤ δ / 2
⊢ False

Next tactic:
suffices : μ (closedBall 0 (δ/2)) = 0
/-- A general version of **Dirichlet's approximation theorem**.

See also `AddCircle.exists_norm_nsmul_le`. -/
lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
    [NormedAddCommGroup A] [CompactSpace A] [ConnectedSpace A]
    [MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
    (ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ/2))) :
    ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ δ := by
  have : IsFiniteMeasure μ := CompactSpace.isFiniteMeasure
  let B : Icc 0 n → Set A := fun j ↦ closedBall ((j : ℕ) • ξ) (δ/2)
  have hB : ∀ j, IsClosed (B j) := fun j ↦ isClosed_ball
  suffices : ¬ Pairwise (Disjoint on B)
  · obtain ⟨i, j, hij, x, hx⟩ := exists_lt_mem_inter_of_not_pairwise_disjoint this
    refine' ⟨j - i, ⟨le_tsub_of_add_le_left hij, _⟩, _⟩
    · simpa only [tsub_le_iff_right] using j.property.2.trans le_self_add
    · rw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), ← sub_eq_add_neg, ← dist_eq_norm]
      refine' (dist_triangle (↑j • ξ) x (↑i • ξ)).trans _
      linarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]
  by_contra h
  apply hn.ne'
  have h' : ⋃ j, B j = univ := by
    rw [← (isClosed_iUnion_of_finite hB).measure_eq_univ_iff_eq (μ := μ)]
    refine' le_antisymm (μ.mono (subset_univ _)) _
    simp_rw [measure_iUnion h (fun _ ↦ measurableSet_closedBall), tsum_fintype,
      μ.addHaar_closedBall_center, Finset.sum_const, Finset.card_univ, Nat.card_fintypeIcc,
      tsub_zero]
    exact hδ
  replace hδ : 0 ≤ δ/2 := by
    by_contra contra
    suffices : μ (closedBall 0 (δ/2)) = 0
    · 

Proof state:
T : ℝ
hT : Fact (0 < T)
A : Type u_1
inst✝⁵ : NormedAddCommGroup A
inst✝⁴ : CompactSpace A
inst✝³ : ConnectedSpace A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure A
inst✝ : Measure.IsAddHaarMeasure μ
ξ : A
n : ℕ
hn : 0 < n
δ : ℝ
hδ : ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))
this✝ : IsFiniteMeasure μ
B : ↑(Icc 0 n) → Set A := fun j => closedBall (↑j • ξ) (δ / 2)
hB : ∀ (j : ↑(Icc 0 n)), IsClosed (B j)
h : Pairwise (Disjoint on B)
h' : ⋃ j, B j = univ
contra : ¬0 ≤ δ / 2
this : ↑↑μ (closedBall 0 (δ / 2)) = 0
⊢ False

Next tactic:
apply isOpen_univ.measure_ne_zero μ univ_nonempty $ le_zero_iff.mp $ le_trans hδ _
/-- A general version of **Dirichlet's approximation theorem**.

See also `AddCircle.exists_norm_nsmul_le`. -/
lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
    [NormedAddCommGroup A] [CompactSpace A] [ConnectedSpace A]
    [MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
    (ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ/2))) :
    ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ δ := by
  have : IsFiniteMeasure μ := CompactSpace.isFiniteMeasure
  let B : Icc 0 n → Set A := fun j ↦ closedBall ((j : ℕ) • ξ) (δ/2)
  have hB : ∀ j, IsClosed (B j) := fun j ↦ isClosed_ball
  suffices : ¬ Pairwise (Disjoint on B)
  · obtain ⟨i, j, hij, x, hx⟩ := exists_lt_mem_inter_of_not_pairwise_disjoint this
    refine' ⟨j - i, ⟨le_tsub_of_add_le_left hij, _⟩, _⟩
    · simpa only [tsub_le_iff_right] using j.property.2.trans le_self_add
    · rw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), ← sub_eq_add_neg, ← dist_eq_norm]
      refine' (dist_triangle (↑j • ξ) x (↑i • ξ)).trans _
      linarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]
  by_contra h
  apply hn.ne'
  have h' : ⋃ j, B j = univ := by
    rw [← (isClosed_iUnion_of_finite hB).measure_eq_univ_iff_eq (μ := μ)]
    refine' le_antisymm (μ.mono (subset_univ _)) _
    simp_rw [measure_iUnion h (fun _ ↦ measurableSet_closedBall), tsum_fintype,
      μ.addHaar_closedBall_center, Finset.sum_const, Finset.card_univ, Nat.card_fintypeIcc,
      tsub_zero]
    exact hδ
  replace hδ : 0 ≤ δ/2 := by
    by_contra contra
    suffices : μ (closedBall 0 (δ/2)) = 0
    · apply isOpen_univ.measure_ne_zero μ univ_nonempty $ le_zero_iff.mp $ le_trans hδ _
      

Proof state:
T : ℝ
hT : Fact (0 < T)
A : Type u_1
inst✝⁵ : NormedAddCommGroup A
inst✝⁴ : CompactSpace A
inst✝³ : ConnectedSpace A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure A
inst✝ : Measure.IsAddHaarMeasure μ
ξ : A
n : ℕ
hn : 0 < n
δ : ℝ
hδ : ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))
this✝ : IsFiniteMeasure μ
B : ↑(Icc 0 n) → Set A := fun j => closedBall (↑j • ξ) (δ / 2)
hB : ∀ (j : ↑(Icc 0 n)), IsClosed (B j)
h : Pairwise (Disjoint on B)
h' : ⋃ j, B j = univ
contra : ¬0 ≤ δ / 2
this : ↑↑μ (closedBall 0 (δ / 2)) = 0
⊢ (n + 1) • ↑↑μ (closedBall 0 (δ / 2)) ≤ 0

Next tactic:
simp [this]
/-- A general version of **Dirichlet's approximation theorem**.

See also `AddCircle.exists_norm_nsmul_le`. -/
lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
    [NormedAddCommGroup A] [CompactSpace A] [ConnectedSpace A]
    [MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
    (ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ/2))) :
    ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ δ := by
  have : IsFiniteMeasure μ := CompactSpace.isFiniteMeasure
  let B : Icc 0 n → Set A := fun j ↦ closedBall ((j : ℕ) • ξ) (δ/2)
  have hB : ∀ j, IsClosed (B j) := fun j ↦ isClosed_ball
  suffices : ¬ Pairwise (Disjoint on B)
  · obtain ⟨i, j, hij, x, hx⟩ := exists_lt_mem_inter_of_not_pairwise_disjoint this
    refine' ⟨j - i, ⟨le_tsub_of_add_le_left hij, _⟩, _⟩
    · simpa only [tsub_le_iff_right] using j.property.2.trans le_self_add
    · rw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), ← sub_eq_add_neg, ← dist_eq_norm]
      refine' (dist_triangle (↑j • ξ) x (↑i • ξ)).trans _
      linarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]
  by_contra h
  apply hn.ne'
  have h' : ⋃ j, B j = univ := by
    rw [← (isClosed_iUnion_of_finite hB).measure_eq_univ_iff_eq (μ := μ)]
    refine' le_antisymm (μ.mono (subset_univ _)) _
    simp_rw [measure_iUnion h (fun _ ↦ measurableSet_closedBall), tsum_fintype,
      μ.addHaar_closedBall_center, Finset.sum_const, Finset.card_univ, Nat.card_fintypeIcc,
      tsub_zero]
    exact hδ
  replace hδ : 0 ≤ δ/2 := by
    by_contra contra
    suffices : μ (closedBall 0 (δ/2)) = 0
    · apply isOpen_univ.measure_ne_zero μ univ_nonempty $ le_zero_iff.mp $ le_trans hδ _
      simp [this]
    

Proof state:
case this
T : ℝ
hT : Fact (0 < T)
A : Type u_1
inst✝⁵ : NormedAddCommGroup A
inst✝⁴ : CompactSpace A
inst✝³ : ConnectedSpace A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure A
inst✝ : Measure.IsAddHaarMeasure μ
ξ : A
n : ℕ
hn : 0 < n
δ : ℝ
hδ : ↑↑μ univ ≤ (n + 1) • ↑↑μ (closedBall 0 (δ / 2))
this : IsFiniteMeasure μ
B : ↑(Icc 0 n) → Set A := fun j => closedBall (↑j • ξ) (δ / 2)
hB : ∀ (j : ↑(Icc 0 n)), IsClosed (B j)
h : Pairwise (Disjoint on B)
h' : ⋃ j, B j = univ
contra : ¬0 ≤ δ / 2
⊢ ↑↑μ (closedBall 0 (δ / 2)) = 0

Next tactic:
rw [not_le, ← closedBall_eq_empty (x := (0 : A))] at contra