awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
s : Set X
r : ℝ≥0∞
h : r > 0
t : ℕ → Set X
ht : s ⊆ iUnion t
n : ℕ
hn : r < ediam (t n)
neg : ¬ediam (t n) ≤ r
neg₁ : (t n).Nonempty
⊢ Eq (α := ℝ≥0∞) (⨆ (_ : (t n).Nonempty), ⨅ (_ : ediam (t n) ≤ r), m (ediam (t n))) ∞

Tactic:
rw [iSup_eq_if, if_pos, iInf_eq_if, if_neg]

Premises used: if_pos, iSup_eq_if, iInf_eq_if, if_neg
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
s : Set X
r : ℝ≥0∞
h : r > 0
t : ℕ → Set X
ht : s ⊆ iUnion t
n : ℕ
hn : r < ediam (t n)
⊢ ¬ediam (t n) ≤ r

Tactic:
hn.not_ge

Premises used: LT.lt.not_ge
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
s : Set X
r : ℝ≥0∞
h : r > 0
t : ℕ → Set X
ht : s ⊆ iUnion t
n : ℕ
hn : r < ediam (t n)
⊢ ¬ediam (t n) ≤ r

Tactic:
exact hn.not_ge

Premises used: LT.lt.not_ge
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
s : Set X
r : ℝ≥0∞
h : r > 0
t : ℕ → Set X
ht : s ⊆ iUnion t
n : ℕ
hn : r < ediam (t n)
neg : ∀ (x : X) (hx : x ∈ t n), (t n).Nonempty
⊢ (t n).Nonempty

Tactic:
rcases ediam_pos_iff.1 hn.pos with ⟨x, hx, -⟩

Premises used: LT.lt.pos, Metric.ediam_pos_iff, Iff.mp
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
s : Set X
r : ℝ≥0∞
h : r > 0
t : ℕ → Set X
ht : s ⊆ iUnion t
n : ℕ
hn : r < ediam (t n)
x : X
hx : x ∈ t n
⊢ (t n).Nonempty

Tactic:
⟨x, hx⟩

Premises used: Set.Nonempty, Exists.intro
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
s : Set X
r : ℝ≥0∞
h : r > 0
t : ℕ → Set X
ht : s ⊆ iUnion t
n : ℕ
hn : r < ediam (t n)
x : X
hx : x ∈ t n
⊢ (t n).Nonempty

Tactic:
exact ⟨x, hx⟩
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
s : Set X
We :
  Eq (α := ℝ≥0∞)
    (⨆ (i : ℝ≥0∞),
      ⨆ (_ : i > 0),
        ⨅ (t : ℕ → Set X),
          ⨅ (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ⨅ (_ : ediam (t n) ≤ i), m (ediam (t n)))
    (⨆ (r : ℝ≥0∞),
      ⨆ (_ : 0 < r),
        ⨅ (t : ℕ → Set X),
          ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)))
We₁ :
  ∀ r > 0,
    ∀ (t : ℕ → Set X) (ht : s ⊆ iUnion t),
      Eq (α := ℝ≥0∞) (∑' (n : ℕ), ⨆ (_ : (id t n).Nonempty), ⨅ (_ : ediam (id t n) ≤ r), m (ediam (id t n)))
        (⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ id r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)))
We₂ :
  ∀ r > 0,
    ∀ (t : ℕ → Set X) (ht : s ⊆ iUnion t),
      Eq (α := ℝ≥0∞) (∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ⨅ (_ : ediam (t n) ≤ r), m (ediam (t n)))
        (⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)))
pos :
  ∀ r > 0,
    ∀ (t : ℕ → Set X) (ht : s ⊆ iUnion t) (htr : ∀ (n : ℕ), ediam (t n) ≤ r),
      Eq (α := ℝ≥0∞) (∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ⨅ (_ : ediam (t n) ≤ r), m (ediam (t n)))
        (⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)))
neg neg₁ :
  ∀ r > 0,
    ∀ (t : ℕ → Set X) (ht : s ⊆ iUnion t) (htr : ¬∀ (n : ℕ), ediam (t n) ≤ r),
      Eq (α := ℝ≥0∞) (∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ⨅ (_ : ediam (t n) ≤ r), m (ediam (t n)))
        (⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)))
⊢ Eq (α := ℝ≥0∞) (((mkMetric m : Measure X) : (a : Set X) → ℝ≥0∞) s : ℝ≥0∞)
    (⨆ (r : ℝ≥0∞),
        ⨆ (_ : 0 < r),
          ⨅ (t : ℕ → Set X),
            ⨅ (_ : s ⊆ iUnion t),
              ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)) :
      ℝ≥0∞)

Tactic:
classical
  -- We mostly unfold the definitions but we need to switch the order of `∑'` and `⨅`
  simp only [← OuterMeasure.coe_mkMetric, OuterMeasure.mkMetric, OuterMeasure.mkMetric',
    OuterMeasure.iSup_apply, OuterMeasure.mkMetric'.pre, OuterMeasure.boundedBy_apply, extend]
  refine
    surjective_id.iSup_congr id fun r =>
      iSup_congr_Prop Iff.rfl fun _ =>
        surjective_id.iInf_congr _ fun t => iInf_congr_Prop Iff.rfl fun ht => ?_
  dsimp
  by_cases htr : ∀ n, ediam (t n) ≤ r
  · rw [iInf_eq_if, if_pos htr]
    congr 1 with n : 1
    simp only [iInf_eq_if, htr n, if_true]
  · rw [iInf_eq_if, if_neg htr]
    push Not at htr; rcases htr with ⟨n, hn⟩
    refine ENNReal.tsum_eq_top_of_eq_top ⟨n, ?_⟩
    rw [iSup_eq_if, if_pos, iInf_eq_if, if_neg]
    · exact hn.not_ge
    rcases ediam_pos_iff.1 hn.pos with ⟨x, hx, -⟩
    exact ⟨x, hx⟩

Premises used: if_true, MeasureTheory.extend, if_pos, MeasureTheory.OuterMeasure.mkMetric', MeasureTheory.OuterMeasure.boundedBy_apply, MeasureTheory.OuterMeasure.mkMetric, MeasureTheory.OuterMeasure.mkMetric'.pre, iSup_eq_if, MeasureTheory.OuterMeasure.coe_mkMetric, iInf_eq_if, MeasureTheory.OuterMeasure.iSup_apply, if_neg, Not
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
μ : Measure X
ε : ℝ≥0∞
h₀ : 0 < ε
h : ∀ (s : Set X), ediam s ≤ ε → (μ : (a : Set X) → ℝ≥0∞) s ≤ m (ediam s)
rw : μ.toOuterMeasure ≤ OuterMeasure.mkMetric m
⊢ μ ≤ mkMetric m

Tactic:
rw [← toOuterMeasure_le, mkMetric_toOuterMeasure]

Premises used: MeasureTheory.Measure.toOuterMeasure_le, MeasureTheory.Measure.mkMetric_toOuterMeasure
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
μ : Measure X
ε : ℝ≥0∞
h₀ : 0 < ε
h : ∀ (s : Set X), ediam s ≤ ε → (μ : (a : Set X) → ℝ≥0∞) s ≤ m (ediam s)
⊢ μ.toOuterMeasure ≤ OuterMeasure.mkMetric m

Tactic:
OuterMeasure.le_mkMetric m μ.toOuterMeasure ε h₀ h

Premises used: MeasureTheory.OuterMeasure.le_mkMetric, MeasureTheory.Measure.toOuterMeasure
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
μ : Measure X
ε : ℝ≥0∞
h₀ : 0 < ε
h : ∀ (s : Set X), ediam s ≤ ε → (μ : (a : Set X) → ℝ≥0∞) s ≤ m (ediam s)
⊢ μ.toOuterMeasure ≤ OuterMeasure.mkMetric m

Tactic:
exact OuterMeasure.le_mkMetric m μ.toOuterMeasure ε h₀ h

Premises used: MeasureTheory.OuterMeasure.le_mkMetric, MeasureTheory.Measure.toOuterMeasure
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
haveI :
  ∀ (this : (n : β) → Encodable (ι n)),
    LE.le (α := ℝ≥0∞) (((mkMetric m : Measure X) : (a : Set X) → ℝ≥0∞) s : ℝ≥0∞)
      (liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) (((mkMetric m : Measure X) : (a : Set X) → ℝ≥0∞) s : ℝ≥0∞)
    (liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l : ℝ≥0∞)

Tactic:
haveI : ∀ n, Encodable (ι n) := fun n => Encodable.ofCountable _
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
simp :
  LE.le (α := ℝ≥0∞)
    (⨆ (r : ℝ≥0∞),
      ⨆ (_ : 0 < r),
        ⨅ (t : ℕ → Set X),
          ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)))
    (liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l)
⊢ LE.le (α := ℝ≥0∞) (((mkMetric m : Measure X) : (a : Set X) → ℝ≥0∞) s : ℝ≥0∞)
    (liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l : ℝ≥0∞)

Tactic:
simp only [mkMetric_apply]

Premises used: MeasureTheory.Measure.mkMetric_apply
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
⊢ LE.le (α := ℝ≥0∞)
    (⨆ (r : ℝ≥0∞),
      ⨆ (_ : 0 < r),
        ⨅ (t : ℕ → Set X),
          ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)))
    (liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l)

Tactic:
iSup₂_le fun ε hε => ?_

Premises used: iSup₂_le
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
refine :
  ∀ (ε : ℝ≥0∞) (hε : 0 < ε),
    LE.le (α := ℝ≥0∞)
      (⨅ (t : ℕ → Set X),
        ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ ε), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)))
      (liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l)
⊢ LE.le (α := ℝ≥0∞)
    (⨆ (r : ℝ≥0∞),
      ⨆ (_ : 0 < r),
        ⨅ (t : ℕ → Set X),
          ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)))
    (liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l)

Tactic:
refine iSup₂_le fun ε hε => ?_

Premises used: iSup₂_le
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
⊢ LE.le (α := ℝ≥0∞)
    (⨅ (t : ℕ → Set X),
      ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ ε), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)))
    (liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l)

Tactic:
le_of_forall_gt_imp_ge_of_dense fun c hc => ?_

Premises used: le_of_forall_gt_imp_ge_of_dense
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
refine :
  ∀ (c : ℝ≥0∞) (hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c),
    ⨅ (t : ℕ → Set X),
        ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ ε), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)) ≤
      c
⊢ LE.le (α := ℝ≥0∞)
    (⨅ (t : ℕ → Set X),
      ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ ε), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)))
    (liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l)

Tactic:
refine le_of_forall_gt_imp_ge_of_dense fun c hc => ?_

Premises used: le_of_forall_gt_imp_ge_of_dense
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
⊢ IsCoboundedUnder (fun (x1 x2 : ℝ≥0∞) ↦ x1 ≥ x2) l fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))

Tactic:
isBoundedDefault
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
rcases :
  ∀ (n : β) (hn : ∑' (i : ι n), m (ediam (t n i)) < c) (hrn : r n < ε) (htn : ∀ (i : ι n), ediam (t n i) ≤ r n)
    (hstn : s ⊆ ⋃ (i : ι n), t n i),
    ⨅ (t : ℕ → Set X),
        ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ ε), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)) ≤
      c
⊢ ⨅ (t : ℕ → Set X),
      ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ ε), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)) ≤
    c

Tactic:
rcases ((frequently_lt_of_liminf_lt (by isBoundedDefault) hc).and_eventually
        ((hr.eventually (gt_mem_nhds hε)).and (ht.and hst))).exists with
    ⟨n, hn, hrn, htn, hstn⟩

Premises used: gt_mem_nhds, Filter.Eventually.and, Filter.Frequently.and_eventually, Filter.Frequently.exists, Filter.frequently_lt_of_liminf_lt, Filter.Tendsto.eventually
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
n : β
hn : ∑' (i : ι n), m (ediam (t n i)) < c
hrn : r n < ε
htn : ∀ (i : ι n), ediam (t n i) ≤ r n
hstn : s ⊆ ⋃ (i : ι n), t n i
u : ℕ → Set X := fun (j : ℕ) ↦ ⋃ (b : ι n), ⋃ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) b), t n b
rwa rwa₁ rwa₂ rwa₃ : s ⊆ ⋃ (b : ι n), t n b
⊢ s ⊆ iUnion u

Tactic:
rwa [iUnion_decode₂]

Premises used: Encodable.iUnion_decode₂
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
n : β
hn : ∑' (i : ι n), m (ediam (t n i)) < c
hrn : r n < ε
htn : ∀ (i : ι n), ediam (t n i) ≤ r n
hstn : s ⊆ ⋃ (i : ι n), t n i
u : ℕ → Set X := fun (j : ℕ) ↦ ⋃ (b : ι n), ⋃ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) b), t n b
refine : ⨅ (_ : ∀ (n : ℕ), ediam (u n) ≤ ε), ∑' (n : ℕ), ⨆ (_ : (u n).Nonempty), m (ediam (u n)) ≤ c
⊢ ⨅ (t : ℕ → Set X),
      ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ ε), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n)) ≤
    c

Tactic:
refine iInf₂_le_of_le u (by rwa [iUnion_decode₂]) ?_

Premises used: iInf₂_le_of_le
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
n : β
hn : ∑' (i : ι n), m (ediam (t n i)) < c
hrn : r n < ε
htn : ∀ (i : ι n), ediam (t n i) ≤ r n
hstn : s ⊆ ⋃ (i : ι n), t n i
u : ℕ → Set X := fun (j : ℕ) ↦ ⋃ (b : ι n), ⋃ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) b), t n b
⊢ ⨅ (_ : ∀ (n : ℕ), ediam (u n) ≤ ε), ∑' (n : ℕ), ⨆ (_ : (u n).Nonempty), m (ediam (u n)) ≤ c

Tactic:
iInf_le_of_le (fun j => ?_) ?_

Premises used: iInf_le_of_le
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
n : β
hn : ∑' (i : ι n), m (ediam (t n i)) < c
hrn : r n < ε
htn : ∀ (i : ι n), ediam (t n i) ≤ r n
hstn : s ⊆ ⋃ (i : ι n), t n i
u : ℕ → Set X := fun (j : ℕ) ↦ ⋃ (b : ι n), ⋃ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) b), t n b
refine_1 : ∀ (j : ℕ), ediam (u j) ≤ ε
refine_2 : ∑' (n : ℕ), ⨆ (_ : (u n).Nonempty), m (ediam (u n)) ≤ c
⊢ ⨅ (_ : ∀ (n : ℕ), ediam (u n) ≤ ε), ∑' (n : ℕ), ⨆ (_ : (u n).Nonempty), m (ediam (u n)) ≤ c

Tactic:
refine iInf_le_of_le (fun j => ?_) ?_

Premises used: iInf_le_of_le
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
n : β
hn : ∑' (i : ι n), m (ediam (t n i)) < c
hrn : r n < ε
htn : ∀ (i : ι n), ediam (t n i) ≤ r n
hstn : s ⊆ ⋃ (i : ι n), t n i
u : ℕ → Set X := fun (j : ℕ) ↦ ⋃ (b : ι n), ⋃ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) b), t n b
j : ℕ
refine_1 : ⨆ (i : ι n), ⨆ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) i), ediam (t n i) ≤ ε
⊢ ediam (u j) ≤ ε

Tactic:
rw [ediam_iUnion_mem_option]

Premises used: Metric.ediam_iUnion_mem_option
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
n : β
hn : ∑' (i : ι n), m (ediam (t n i)) < c
hrn : r n < ε
htn : ∀ (i : ι n), ediam (t n i) ≤ r n
hstn : s ⊆ ⋃ (i : ι n), t n i
u : ℕ → Set X := fun (j : ℕ) ↦ ⋃ (b : ι n), ⋃ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) b), t n b
j : ℕ
⊢ ⨆ (i : ι n), ⨆ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) i), ediam (t n i) ≤ ε

Tactic:
iSup₂_le fun _ _ => (htn _).trans hrn.le

Premises used: iSup₂_le, LT.lt.le, LE.le.trans
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
n : β
hn : ∑' (i : ι n), m (ediam (t n i)) < c
hrn : r n < ε
htn : ∀ (i : ι n), ediam (t n i) ≤ r n
hstn : s ⊆ ⋃ (i : ι n), t n i
u : ℕ → Set X := fun (j : ℕ) ↦ ⋃ (b : ι n), ⋃ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) b), t n b
j : ℕ
⊢ ⨆ (i : ι n), ⨆ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) i), ediam (t n i) ≤ ε

Tactic:
exact iSup₂_le fun _ _ => (htn _).trans hrn.le

Premises used: iSup₂_le, LT.lt.le, LE.le.trans
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
n : β
hn : ∑' (i : ι n), m (ediam (t n i)) < c
hrn : r n < ε
htn : ∀ (i : ι n), ediam (t n i) ≤ r n
hstn : s ⊆ ⋃ (i : ι n), t n i
u : ℕ → Set X := fun (j : ℕ) ↦ ⋃ (b : ι n), ⋃ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) b), t n b
⊢ Eq (α := ℝ≥0∞) ((fun (t : Set X) ↦ (⨆ (_ : t.Nonempty), m (ediam t) : ℝ≥0∞)) ∅) 0

Tactic:
simp
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
n : β
hn : ∑' (i : ι n), m (ediam (t n i)) < c
hrn : r n < ε
htn : ∀ (i : ι n), ediam (t n i) ≤ r n
hstn : s ⊆ ⋃ (i : ι n), t n i
u : ℕ → Set X := fun (j : ℕ) ↦ ⋃ (b : ι n), ⋃ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) b), t n b
⊢ LE.le (α := ℝ≥0∞) (∑' (b : ι n), ⨆ (_ : (t n b).Nonempty), m (ediam (t n b))) (∑' (i : ι n), m (ediam (t n i)))

Tactic:
ENNReal.tsum_le_tsum fun b => iSup_le fun _ => le_rfl

Premises used: ENNReal.tsum_le_tsum, le_rfl, iSup_le
-- Module: Mathlib.MeasureTheory.Measure.Hausdorff
Goal:
X : Type u_2
inst : EMetricSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst_3 : ∀ (n : β), Countable.{u_5 + 1} (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ (i : ι n), t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun (n : β) ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
n : β
hn : ∑' (i : ι n), m (ediam (t n i)) < c
hrn : r n < ε
htn : ∀ (i : ι n), ediam (t n i) ≤ r n
hstn : s ⊆ ⋃ (i : ι n), t n i
u : ℕ → Set X := fun (j : ℕ) ↦ ⋃ (b : ι n), ⋃ (_ : Membership.mem (γ := Option (ι n)) (decode₂ (ι n) j) b), t n b
⊢ ∑' (i : ι n), m (ediam (t n i)) ≤ c

Tactic:
hn.le

Premises used: LT.lt.le