awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : ∀ (x : 𝓧), ∀ r > 0, ∃ (y : ℕ), dist x (D y) < r
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
m : ℕ
p : 𝓧
⊢ Membership.mem (γ := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) p ↔ Membership.mem (γ := Set 𝓧) univ p

Tactic:
⟨fun a ↦ trivial, fun _ ↦ mem_iUnion.mpr <| hD p (u m) (hu_pos m)⟩

Premises used: Iff.mpr, trivial, Set.mem_iUnion, Iff.intro
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : ∀ (x : 𝓧), ∀ r > 0, ∃ (y : ℕ), dist x (D y) < r
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
m : ℕ
p : 𝓧
⊢ Membership.mem (γ := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) p ↔ Membership.mem (γ := Set 𝓧) univ p

Tactic:
exact ⟨fun a ↦ trivial, fun _ ↦ mem_iUnion.mpr <| hD p (u m) (hu_pos m)⟩
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
⊢ ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ

Tactic:
have hcov (m : ℕ) : ⋃ i, ball (D i) (u m) = univ := by
    rw [denseRange_iff] at hD
    ext p
    exact ⟨fun a ↦ trivial, fun _ ↦ mem_iUnion.mpr <| hD p (u m) (hu_pos m)⟩
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
inr :
  ∀ (hεbound : 1 < ε),
    ∃ (K : Set 𝓧),
      IsCompact K ∧
        ∀ (μ : Measure 𝓧),
          Membership.mem (γ := Set (Measure 𝓧)) {x : Measure 𝓧 | ∃ μ ∈ S, (↑μ : Measure 𝓧) = x} μ →
            (μ : (a : Set 𝓧) → ℝ≥0∞) Kᶜ ≤ ε
inr₁ :
  ∀ (hεbound : ε ≤ 1),
    ∃ (K : Set 𝓧),
      IsCompact K ∧
        ∀ (μ : Measure 𝓧),
          Membership.mem (γ := Set (Measure 𝓧)) {x : Measure 𝓧 | ∃ μ ∈ S, (↑μ : Measure 𝓧) = x} μ →
            (μ : (a : Set 𝓧) → ℝ≥0∞) Kᶜ ≤ ε
⊢ ∃ (K : Set 𝓧),
    IsCompact K ∧
      ∀ (μ : Measure 𝓧),
        Membership.mem (γ := Set (Measure 𝓧)) {x : Measure 𝓧 | ∃ μ ∈ S, (↑μ : Measure 𝓧) = x} μ →
          (μ : (a : Set 𝓧) → ℝ≥0∞) Kᶜ ≤ ε

Tactic:
rcases lt_or_ge 1 ε with hεbound | hεbound

Premises used: lt_or_ge, OfNat.ofNat
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : 1 < ε
⊢ ∃ (K : Set 𝓧),
    IsCompact K ∧
      ∀ (μ : Measure 𝓧),
        Membership.mem (γ := Set (Measure 𝓧)) {x : Measure 𝓧 | ∃ μ ∈ S, (↑μ : Measure 𝓧) = x} μ →
          (μ : (a : Set 𝓧) → ℝ≥0∞) Kᶜ ≤ ε

Tactic:
⟨∅, isCompact_empty, fun μ hμ ↦ ?_⟩

Premises used: And.intro, Exists.intro, EmptyCollection.emptyCollection, isCompact_empty
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : 1 < ε
inr :
  ∀ (μ : Measure 𝓧) (hμ : Membership.mem (γ := Set (Measure 𝓧)) {x : Measure 𝓧 | ∃ μ ∈ S, (↑μ : Measure 𝓧) = x} μ),
    (μ : (a : Set 𝓧) → ℝ≥0∞) ∅ᶜ ≤ ε
⊢ ∃ (K : Set 𝓧),
    IsCompact K ∧
      ∀ (μ : Measure 𝓧),
        Membership.mem (γ := Set (Measure 𝓧)) {x : Measure 𝓧 | ∃ μ ∈ S, (↑μ : Measure 𝓧) = x} μ →
          (μ : (a : Set 𝓧) → ℝ≥0∞) Kᶜ ≤ ε

Tactic:
refine ⟨∅, isCompact_empty, fun μ hμ ↦ ?_⟩
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : 1 < ε
μ : Measure 𝓧
hμ : ∃ μ_1 ∈ S, (↑μ_1 : Measure 𝓧) = μ
inr : ∀ (μ' : ProbabilityMeasure 𝓧) (hμ' : μ' ∈ S), ((↑μ' : Measure 𝓧) : (a : Set 𝓧) → ℝ≥0∞) ∅ᶜ ≤ ε
⊢ (μ : (a : Set 𝓧) → ℝ≥0∞) ∅ᶜ ≤ ε

Tactic:
obtain ⟨μ', hμ', rfl⟩ := hμ

Premises used: rfl
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : 1 < ε
μ' : ProbabilityMeasure 𝓧
hμ' : μ' ∈ S
inr : 1 ≤ ε
⊢ ((↑μ' : Measure 𝓧) : (a : Set 𝓧) → ℝ≥0∞) ∅ᶜ ≤ ε

Tactic:
rw [compl_empty, measure_univ]

Premises used: MeasureTheory.IsProbabilityMeasure.measure_univ, Set.compl_empty
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : 1 < ε
μ' : ProbabilityMeasure 𝓧
hμ' : μ' ∈ S
⊢ 1 ≤ ε

Tactic:
le_of_lt hεbound

Premises used: le_of_lt
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : 1 < ε
μ' : ProbabilityMeasure 𝓧
hμ' : μ' ∈ S
⊢ 1 ≤ ε

Tactic:
exact le_of_lt hεbound

Premises used: le_of_lt
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
m : ℕ
⊢ ∃ (k : ℕ),
    ∀ μ ∈ S,
      1 - ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ)) <
        (↑((μ : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), ball (D i) (u m))) : ℝ≥0∞)

Tactic:
exists_measure_iUnion_gt_of_isCompact_closure
      (fun i ↦ ball (D i) (u m)) (fun _ ↦ isOpen_ball) (hcov m) hcomp (ε * 2 ^ (-m : ℤ)) ?_ ?_

Premises used: HMul.hMul, Metric.ball, Nat.cast, Metric.isOpen_ball, HPow.hPow, MeasureTheory.exists_measure_iUnion_gt_of_isCompact_closure, OfNat.ofNat, Neg.neg
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
m : ℕ
refine_1 : 0 < ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ))
refine_2 : ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ)) ≤ 1
⊢ ∃ (k : ℕ),
    ∀ μ ∈ S,
      1 - ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ)) <
        (↑((μ : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), ball (D i) (u m))) : ℝ≥0∞)

Tactic:
refine exists_measure_iUnion_gt_of_isCompact_closure
      (fun i ↦ ball (D i) (u m)) (fun _ ↦ isOpen_ball) (hcov m) hcomp (ε * 2 ^ (-m : ℤ)) ?_ ?_

Premises used: HMul.hMul, Metric.ball, Nat.cast, Metric.isOpen_ball, HPow.hPow, MeasureTheory.exists_measure_iUnion_gt_of_isCompact_closure, OfNat.ofNat, Neg.neg
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
m : ℕ
⊢ Ne (α := ℝ≥0∞) 2 0

Tactic:
simp
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
m : ℕ
⊢ Ne (α := ℝ≥0∞) 2 ∞

Tactic:
simp
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
m : ℕ
⊢ 0 < ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ))

Tactic:
simpa using ⟨εpos, (ENNReal.zpow_pos (by simp) (by simp) (-↑m))⟩
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
m : ℕ
linarith : LT.lt (α := ℤ) 0 (-(↑m : ℤ)) → -1 + HSub.hSub (α := ℤ) 0 (↑m : ℤ) + (HAdd.hAdd (α := ℤ) 0 1 - -(↑m : ℤ)) = 0
⊢ LE.le (α := ℤ) (-(↑m : ℤ)) 0

Tactic:
linarith
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
m : ℕ
⊢ LE.le (α := ℝ≥0∞) 1 2

Tactic:
simp
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
m : ℕ
⊢ ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ)) ≤ 1

Tactic:
exact Left.mul_le_one hεbound <| zpow_le_one_of_nonpos (by linarith) (by simp)

Premises used: Left.mul_le_one, ENNReal.zpow_le_one_of_nonpos
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
⊢ ∀ (m : ℕ),
    ∃ (k : ℕ),
      ∀ μ ∈ S,
        1 - ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ)) <
          (↑((μ : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), ball (D i) (u m))) : ℝ≥0∞)

Tactic:
have byclaim (m : ℕ) : ∃ k, ∀ μ ∈ S, 1 - (ε * 2 ^ (-m : ℤ) : ℝ≥0∞) <
      μ (⋃ i ≤ k, ball (D i) (u m)) := by
    refine exists_measure_iUnion_gt_of_isCompact_closure
      (fun i ↦ ball (D i) (u m)) (fun _ ↦ isOpen_ball) (hcov m) hcomp (ε * 2 ^ (-m : ℤ)) ?_ ?_
    · simpa using ⟨εpos, (ENNReal.zpow_pos (by simp) (by simp) (-↑m))⟩
    · exact Left.mul_le_one hεbound <| zpow_le_one_of_nonpos (by linarith) (by simp)
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
byclaim :
  ∀ (m : ℕ),
    ∃ (k : ℕ),
      ∀ μ ∈ S,
        1 - ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ)) <
          (↑((μ : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), ball (D i) (u m))) : ℝ≥0∞)
inr :
  ∀ (km : (m : ℕ) → ℕ)
    (hbound :
      ∀ (m : ℕ),
        ∀ μ ∈ S,
          1 - ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ)) <
            (↑((μ : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ km m), ball (D i) (u m))) : ℝ≥0∞)),
    ∃ (K : Set 𝓧),
      IsCompact K ∧
        ∀ (μ : Measure 𝓧),
          Membership.mem (γ := Set (Measure 𝓧)) {x : Measure 𝓧 | ∃ μ ∈ S, (↑μ : Measure 𝓧) = x} μ →
            (μ : (a : Set 𝓧) → ℝ≥0∞) Kᶜ ≤ ε
⊢ ∃ (K : Set 𝓧),
    IsCompact K ∧
      ∀ (μ : Measure 𝓧),
        Membership.mem (γ := Set (Measure 𝓧)) {x : Measure 𝓧 | ∃ μ ∈ S, (↑μ : Measure 𝓧) = x} μ →
          (μ : (a : Set 𝓧) → ℝ≥0∞) Kᶜ ≤ ε

Tactic:
choose! km hbound using byclaim
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
km : (m : ℕ) → ℕ
hbound :
  ∀ (m : ℕ),
    ∀ μ ∈ S,
      1 - ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ)) <
        (↑((μ : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ km m), ball (D i) (u m))) : ℝ≥0∞)
bigK : Set 𝓧 :=
  ⋂ (m : ℕ),
    ⋃ (i : ℕ), ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))
μ : ProbabilityMeasure 𝓧
hs : μ ∈ S
⊢ Eq (α := ℝ≥0∞) (((↑μ : Measure 𝓧) : (a : Set 𝓧) → ℝ≥0∞) bigKᶜ : ℝ≥0∞)
    (((↑μ : Measure 𝓧) : (a : Set 𝓧) → ℝ≥0∞)
        (⋃ (m : ℕ),
          (⋃ (i : ℕ),
              ⋃ (_ : i ≤ km (m + 1)),
                closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m)))ᶜ) :
      ℝ≥0∞)

Tactic:
simp [bigK]
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
km : (m : ℕ) → ℕ
hbound :
  ∀ (m : ℕ),
    ∀ μ ∈ S,
      1 - ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ)) <
        (↑((μ : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ km m), ball (D i) (u m))) : ℝ≥0∞)
bigK : Set 𝓧 :=
  ⋂ (m : ℕ),
    ⋃ (i : ℕ), ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))
μ : ProbabilityMeasure 𝓧
hs : μ ∈ S
h :
  ∀ (m : ℕ),
    ((↑μ : Measure 𝓧) : (a : Set 𝓧) → ℝ≥0∞)
        (⋃ (i : ℕ),
            ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m)))ᶜ =
      HSub.hSub (α := ℝ≥0∞) (β := ℝ≥0∞) (1 : ℝ≥0∞)
        (((↑μ : Measure 𝓧) : (a : Set 𝓧) → ℝ≥0∞)
            (⋃ (i : ℕ),
              ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))) :
          ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞)
    (∑' (m : ℕ),
      ((↑μ : Measure 𝓧) : (a : Set 𝓧) → ℝ≥0∞)
        (⋃ (i : ℕ),
            ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m)))ᶜ)
    (∑' (m : ℕ),
      HSub.hSub (α := ℝ≥0∞) (β := ℝ≥0∞) (1 : ℝ≥0∞)
        (((↑μ : Measure 𝓧) : (a : Set 𝓧) → ℝ≥0∞)
            (⋃ (i : ℕ),
              ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))) :
          ℝ≥0∞))

Tactic:
congr! with m
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
km : (m : ℕ) → ℕ
hbound :
  ∀ (m : ℕ),
    ∀ μ ∈ S,
      1 - ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ)) <
        (↑((μ : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ km m), ball (D i) (u m))) : ℝ≥0∞)
bigK : Set 𝓧 :=
  ⋂ (m : ℕ),
    ⋃ (i : ℕ), ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))
μ : ProbabilityMeasure 𝓧
hs : μ ∈ S
m : ℕ
⊢ MeasurableSet.{u_1} (α := 𝓧)
    (⋃ (i : ℕ), ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m)))

Tactic:
measurability
-- Module: Mathlib.MeasureTheory.Measure.Tight
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst : PseudoMetricSpace 𝓧
inst_1 : OpensMeasurableSpace 𝓧
inst_2 : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst_3 : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
km : (m : ℕ) → ℕ
hbound :
  ∀ (m : ℕ),
    ∀ μ ∈ S,
      1 - ε * HPow.hPow (α := ℝ≥0∞) (β := ℤ) 2 (-(↑m : ℤ)) <
        (↑((μ : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ km m), ball (D i) (u m))) : ℝ≥0∞)
bigK : Set 𝓧 :=
  ⋂ (m : ℕ),
    ⋃ (i : ℕ), ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))
μ : ProbabilityMeasure 𝓧
hs : μ ∈ S
m : ℕ
⊢ Ne (α := ℝ≥0∞)
    (((↑μ : Measure 𝓧) : (a : Set 𝓧) → ℝ≥0∞)
        (⋃ (i : ℕ),
          ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))) :
      ℝ≥0∞)
    (∞ : ℝ≥0∞)

Tactic:
simp