awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
this : ∀ (n : ℕ), S n ⊆ S (n + 1)
neg : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (k + 1) \ S k)) ∞
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (⋃ (n : ℕ), S n) : ℝ≥0∞)
    (⨆ (n : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞)

Tactic:
refine (μ.iUnion_nat_of_monotone_of_tsum_ne_top this ?_).le

Premises used: MeasureTheory.OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top, Eq.le
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
neg :
  Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1) \ S (2 * k))) ∞ ∧
    Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1 + 1) \ S (2 * k + 1))) ∞
⊢ Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (k + 1) \ S k)) ∞

Tactic:
rw [← tsum_even_add_odd ENNReal.summable ENNReal.summable, ENNReal.add_ne_top]

Premises used: ENNReal.summable, tsum_even_add_odd, ENNReal.add_ne_top
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
this : ∀ (a : ℕ), Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1 + a) \ S (2 * k + a))) ∞
⊢ Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1) \ S (2 * k))) ∞

Tactic:
simpa using this 0

Premises used: OfNat.ofNat
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
this : ∀ (a : ℕ), Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1 + a) \ S (2 * k + a))) ∞
⊢ Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1 + 1) \ S (2 * k + 1))) ∞

Tactic:
simpa using this 1

Premises used: OfNat.ofNat
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
neg : ∀ (a : ℕ), Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1 + a) \ S (2 * k + a))) ∞
⊢ Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1) \ S (2 * k))) ∞ ∧
    Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1 + 1) \ S (2 * k + 1))) ∞

Tactic:
suffices ∀ a, (∑' k : ℕ, μ (S (2 * k + 1 + a) \ S (2 * k + a))) ≠ ∞ from
    ⟨by simpa using this 0, by simpa using this 1⟩
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ ∀ (a : ℕ), Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1 + a) \ S (2 * k + a))) ∞

Tactic:
fun r => ne_top_of_le_ne_top htop ?_

Premises used: ne_top_of_le_ne_top
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
neg :
  ∀ (r : ℕ),
    LE.le (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1 + r) \ S (2 * k + r)) : ℝ≥0∞)
      ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞)
⊢ ∀ (a : ℕ), Ne (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1 + a) \ S (2 * k + a))) ∞

Tactic:
refine fun r => ne_top_of_le_ne_top htop ?_

Premises used: ne_top_of_le_ne_top
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r : ℕ
neg :
  ∀ (i : ℕ),
    LE.le (α := ℝ≥0∞) (∑ a ∈ Finset.range i, (μ : (a : Set X) → ℝ≥0∞) (S (2 * a + 1 + r) \ S (2 * a + r)) : ℝ≥0∞)
      ((μ : (a : Set X) → ℝ≥0∞) (⋃ (n : ℕ), S n) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) (∑' (k : ℕ), (μ : (a : Set X) → ℝ≥0∞) (S (2 * k + 1 + r) \ S (2 * k + r)) : ℝ≥0∞)
    ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞)

Tactic:
rw [← iUnion_S, ENNReal.tsum_eq_iSup_nat, iSup_le_iff]

Premises used: ENNReal.tsum_eq_iSup_nat, iSup_le_iff
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n : ℕ
neg :
  LE.le (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (⋃ i ∈ Finset.range n, S (2 * i + 1 + r) \ S (2 * i + r)) : ℝ≥0∞)
    ((μ : (a : Set X) → ℝ≥0∞) (⋃ (n : ℕ), S n) : ℝ≥0∞)
neg₁ :
  ∀ i ∈ Finset.range n,
    ∀ j ∈ Finset.range n, i ≠ j → AreSeparated (S (2 * i + 1 + r) \ S (2 * i + r)) (S (2 * j + 1 + r) \ S (2 * j + r))
⊢ LE.le (α := ℝ≥0∞) (∑ a ∈ Finset.range n, (μ : (a : Set X) → ℝ≥0∞) (S (2 * a + 1 + r) \ S (2 * a + r)) : ℝ≥0∞)
    ((μ : (a : Set X) → ℝ≥0∞) (⋃ (n : ℕ), S n) : ℝ≥0∞)

Tactic:
rw [← hm.finset_iUnion_of_pairwise_separated]

Premises used: MeasureTheory.OuterMeasure.IsMetric.finset_iUnion_of_pairwise_separated
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n : ℕ
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (⋃ i ∈ Finset.range n, S (2 * i + 1 + r) \ S (2 * i + r)) : ℝ≥0∞)
    ((μ : (a : Set X) → ℝ≥0∞) (⋃ (n : ℕ), S n) : ℝ≥0∞)

Tactic:
μ.mono (iUnion_subset fun i => iUnion_subset fun _ x hx => mem_iUnion.2 ⟨_, hx.1⟩)

Premises used: Iff.mpr, HMul.hMul, Set.iUnion_subset, MeasureTheory.OuterMeasure.mono, Set.mem_iUnion, And.left, HAdd.hAdd, Exists.intro, OfNat.ofNat
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n : ℕ
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (⋃ i ∈ Finset.range n, S (2 * i + 1 + r) \ S (2 * i + r)) : ℝ≥0∞)
    ((μ : (a : Set X) → ℝ≥0∞) (⋃ (n : ℕ), S n) : ℝ≥0∞)

Tactic:
exact μ.mono (iUnion_subset fun i => iUnion_subset fun _ x hx => mem_iUnion.2 ⟨_, hx.1⟩)

Premises used: Iff.mpr, Set.iUnion_subset, MeasureTheory.OuterMeasure.mono, Set.mem_iUnion, Exists.intro
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n : ℕ
this : ∀ (i j : ℕ), i < j → AreSeparated (S (2 * i + 1 + r)) (s \ S (2 * j + r))
i : ℕ
h_1 : i ∈ Finset.range n
j : ℕ
h_2 : j ∈ Finset.range n
hij : i ≠ j
h : i < j
x : X
hx : x ∈ S (2 * j + 1 + r) \ S (2 * j + r)
⊢ x ∈ s \ S (2 * j + r)

Tactic:
⟨hx.1.1, hx.2⟩

Premises used: Set.Mem, Set.diff, setOf, And.right, And.left, And.intro
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n : ℕ
this : ∀ (i j : ℕ), i < j → AreSeparated (S (2 * i + 1 + r)) (s \ S (2 * j + r))
i : ℕ
h_1 : i ∈ Finset.range n
j : ℕ
h_2 : j ∈ Finset.range n
hij : i ≠ j
h : i < j
x : X
hx : x ∈ S (2 * j + 1 + r) \ S (2 * j + r)
⊢ x ∈ s \ S (2 * j + r)

Tactic:
exact ⟨hx.1.1, hx.2⟩
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n : ℕ
this : ∀ (i j : ℕ), i < j → AreSeparated (S (2 * i + 1 + r)) (s \ S (2 * j + r))
i : ℕ
h_1 : i ∈ Finset.range n
j : ℕ
h_2 : j ∈ Finset.range n
hij : i ≠ j
h : j < i
x : X
hx : x ∈ S (2 * i + 1 + r) \ S (2 * i + r)
⊢ x ∈ s \ S (2 * i + r)

Tactic:
⟨hx.1.1, hx.2⟩

Premises used: Set.Mem, Set.diff, setOf, And.right, And.left, And.intro
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n : ℕ
this : ∀ (i j : ℕ), i < j → AreSeparated (S (2 * i + 1 + r)) (s \ S (2 * j + r))
i : ℕ
h_1 : i ∈ Finset.range n
j : ℕ
h_2 : j ∈ Finset.range n
hij : i ≠ j
h : j < i
x : X
hx : x ∈ S (2 * i + 1 + r) \ S (2 * i + r)
⊢ x ∈ s \ S (2 * i + r)

Tactic:
exact ⟨hx.1.1, hx.2⟩
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n : ℕ
neg : ∀ (i j : ℕ), i < j → AreSeparated (S (2 * i + 1 + r)) (s \ S (2 * j + r))
⊢ ∀ i ∈ Finset.range n,
    ∀ j ∈ Finset.range n, i ≠ j → AreSeparated (S (2 * i + 1 + r) \ S (2 * i + r)) (S (2 * j + 1 + r) \ S (2 * j + r))

Tactic:
suffices ∀ i j, i < j → Metric.AreSeparated (S (2 * i + 1 + r)) (s \ S (2 * j + r)) from
    fun i _ j _ hij => hij.lt_or_gt.elim
      (fun h => (this i j h).mono inter_subset_left fun x hx => by exact ⟨hx.1.1, hx.2⟩)
      fun h => (this j i h).symm.mono (fun x hx => by exact ⟨hx.1.1, hx.2⟩) inter_subset_left
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n i j : ℕ
hj : i < j
rw : 2 * i + 1 + r < 2 * j + r
⊢ LT.lt (α := ℝ≥0∞) (↑(2 * j + r) : ℝ≥0∞)⁻¹ (↑(2 * i + 1 + r) : ℝ≥0∞)⁻¹

Tactic:
rw [ENNReal.inv_lt_inv, Nat.cast_lt]

Premises used: Nat.cast_lt, ENNReal.inv_lt_inv
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n i j : ℕ
hj : i < j
⊢ 2 * i + 1 + r < 2 * j + r

Tactic:
lia
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n i j : ℕ
hj : i < j
⊢ LT.lt (α := ℝ≥0∞) (↑(2 * j + r) : ℝ≥0∞)⁻¹ (↑(2 * i + 1 + r) : ℝ≥0∞)⁻¹

Tactic:
have A : ((↑(2 * j + r))⁻¹ : ℝ≥0∞) < (↑(2 * i + 1 + r))⁻¹ := by
    rw [ENNReal.inv_lt_inv, Nat.cast_lt]; lia
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n i j : ℕ
hj : i < j
A : LT.lt (α := ℝ≥0∞) (↑(2 * j + r) : ℝ≥0∞)⁻¹ (↑(2 * i + 1 + r) : ℝ≥0∞)⁻¹
⊢ HSub.hSub (α := ℝ≥0∞) (↑(2 * i + 1 + r) : ℝ≥0∞)⁻¹ (↑(2 * j + r) : ℝ≥0∞)⁻¹ ≠ 0

Tactic:
simpa [tsub_eq_zero_iff_le] using A

Premises used: tsub_eq_zero_iff_le
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n i j : ℕ
hj : i < j
A : LT.lt (α := ℝ≥0∞) (↑(2 * j + r) : ℝ≥0∞)⁻¹ (↑(2 * i + 1 + r) : ℝ≥0∞)⁻¹
neg :
  ∀ (x : X) (hx : x ∈ S (2 * i + 1 + r)) (y : X) (hy : y ∈ s \ S (2 * j + r)),
    HSub.hSub (α := ℝ≥0∞) (↑(2 * i + 1 + r) : ℝ≥0∞)⁻¹ (↑(2 * j + r) : ℝ≥0∞)⁻¹ ≤ edist x y
⊢ AreSeparated (S (2 * i + 1 + r)) (s \ S (2 * j + r))

Tactic:
refine ⟨(↑(2 * i + 1 + r))⁻¹ - (↑(2 * j + r))⁻¹, by simpa [tsub_eq_zero_iff_le] using A,
    fun x hx y hy => ?_⟩
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : Membership.mem (γ := Set (Set X)) {s : Set X | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s} t
s : Set X
S : ℕ → Set X := fun (n : ℕ) ↦ {x : X | x ∈ s ∧ (↑n : ℝ≥0∞)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs :
  ∀ (n : ℕ),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞)
        ((μ : (a : Set X) → ℝ≥0∞) (S n) : ℝ≥0∞) ≤
      (μ : (a : Set X) → ℝ≥0∞) s
iUnion_S : ⋃ (n : ℕ), S n = s \ t
htop : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set X) → ℝ≥0∞) (s \ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
r n i j : ℕ
hj : i < j
A : LT.lt (α := ℝ≥0∞) (↑(2 * j + r) : ℝ≥0∞)⁻¹ (↑(2 * i + 1 + r) : ℝ≥0∞)⁻¹
x : X
hx : x ∈ S (2 * i + 1 + r)
y : X
hy : y ∈ s \ S (2 * j + r)
⊢ infEDist y t < (↑(2 * j + r) : ℝ≥0∞)⁻¹

Tactic:
have : infEDist y t < (↑(2 * j + r))⁻¹ := not_le.1 fun hle => hy.2 ⟨hy.1, hle⟩