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)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), U i) univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : (k : ℕ) → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction :
  ∀ (k : ℕ), (↑((μ k : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), U i)) : ℝ≥0∞) ≤ 1 - (↑(NNReal.mk ε hε') : ℝ≥0∞)
μlim : ProbabilityMeasure 𝓧
left : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
n : ℕ
b : ℝ
c : ℕ
h : ∀ (b_1 : ℕ), c ≤ b_1 → b ≤ (↑((μ (sub b_1) : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ sub b_1), U i)) : ℝ)
h₁ h₂ h₃ h₄ : (↑(1 - NNReal.mk ε hε') : ℝ≥0∞) ≤ (↑(1 - NNReal.mk ε hε') : ℝ≥0∞)
⊢ 1 - (↑(NNReal.mk ε hε') : ℝ≥0∞) ≤ ENNReal.ofReal (1 - ε)

Tactic:
norm_cast
-- 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)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), U i) univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : (k : ℕ) → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction :
  ∀ (k : ℕ), (↑((μ k : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), U i)) : ℝ≥0∞) ≤ 1 - (↑(NNReal.mk ε hε') : ℝ≥0∞)
μlim : ProbabilityMeasure 𝓧
left : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
⊢ ∀ (n : ℕ), (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ) ≤ 1 - ε

Tactic:
have Measurebound n : (μlim (⋃ (i ≤ n), U i) : ℝ) ≤ 1 - ε := calc
    (μlim (⋃ (i ≤ n), U i) : ℝ)
    _ ≤ liminf (fun k ↦ (μ (sub k) (⋃ (i ≤ n), U i) : ℝ)) atTop := by
      have hopen : IsOpen (⋃ i ≤ n, U i) := isOpen_biUnion fun i a ↦ O i
      have := ProbabilityMeasure.le_liminf_measure_open_of_tendsto hμconverges hopen
      simp_rw [Function.comp_apply, ← ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure] at this
      rw [← ofNNReal_liminf] at this
      · exact mod_cast this
      use 1
      simpa [ge_iff_le, eventually_map, eventually_atTop, forall_exists_index] using fun _ x h ↦
          (h x (by simp)).trans <| ProbabilityMeasure.apply_le_one (μ (sub x)) (⋃ i ≤ n, U i)
    _ ≤ liminf (fun k ↦ (μ (sub k) (⋃ (i ≤ sub k), U i) : ℝ)) atTop := by
      apply Filter.liminf_le_liminf
      · simp only [NNReal.coe_le_coe, eventually_atTop, ge_iff_le]
        use n + 1
        intro b hypo
        refine (μ (sub b)).apply_mono
            <| Set.biUnion_mono (fun i (hi : i ≤ n) ↦ hi.trans ?_) fun _ _ ↦ le_rfl
        exact le_trans (Nat.le_add_right n 1) (le_trans hypo (StrictMono.le_apply hsubmono))
      · use 0; simp
      · use 1
        simpa [ge_iff_le, eventually_map, eventually_atTop, ge_iff_le, forall_exists_index] using
            fun _ d hyp ↦ (hyp d (by simp)).trans (by simp)
    _ ≤ 1 - ε := by
      apply Filter.liminf_le_of_le
      · use 0; simp
      simp only [eventually_atTop, ge_iff_le, forall_exists_index]
      intro b c h
      apply le_trans (h c le_rfl)
      refine (ofReal_le_ofReal_iff (by rw [sub_nonneg]; exact hεbound)).mp ?_
      rw [ofReal_coe_nnreal]
      apply le_trans (hcontradiction (sub c))
      norm_cast
-- 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)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), U i) univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : (k : ℕ) → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction :
  ∀ (k : ℕ), (↑((μ k : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), U i)) : ℝ≥0∞) ≤ 1 - (↑(NNReal.mk ε hε') : ℝ≥0∞)
μlim : ProbabilityMeasure 𝓧
left : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
Measurebound : ∀ (n : ℕ), (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ) ≤ 1 - ε
simp_rw :
  Tendsto (β := ℝ≥0) (fun (n : ℕ) ↦ ((μlim : (a : Set 𝓧) → ℝ≥0) (accumulate U n) : ℝ≥0)) atTop
    (𝓝 ((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), U i)) : Filter ℝ≥0)
⊢ Tendsto (β := ℝ≥0) (fun (n : ℕ) ↦ ((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i) : ℝ≥0)) atTop
    (𝓝 ((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), U i)) : Filter ℝ≥0)

Tactic:
simp_rw [← Set.accumulate_def, ProbabilityMeasure.tendsto_measure_iUnion_accumulate]

Premises used: MeasureTheory.ProbabilityMeasure.tendsto_measure_iUnion_accumulate, Set.accumulate_def
-- 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)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), U i) univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : (k : ℕ) → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction :
  ∀ (k : ℕ), (↑((μ k : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), U i)) : ℝ≥0∞) ≤ 1 - (↑(NNReal.mk ε hε') : ℝ≥0∞)
μlim : ProbabilityMeasure 𝓧
left : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
Measurebound : ∀ (n : ℕ), (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ) ≤ 1 - ε
⊢ Tendsto (β := ℝ≥0) (fun (n : ℕ) ↦ ((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i) : ℝ≥0)) atTop
    (𝓝 ((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), U i)) : Filter ℝ≥0)

Tactic:
have accumulation : Tendsto (fun n ↦ μlim (⋃ i ≤ n, U i)) atTop (𝓝 (μlim (⋃ i, U i))) := by
    simp_rw [← Set.accumulate_def, ProbabilityMeasure.tendsto_measure_iUnion_accumulate]
-- 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)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), U i) univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : (k : ℕ) → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction :
  ∀ (k : ℕ), (↑((μ k : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), U i)) : ℝ≥0∞) ≤ 1 - (↑(NNReal.mk ε hε') : ℝ≥0∞)
μlim : ProbabilityMeasure 𝓧
left : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
Measurebound : ∀ (n : ℕ), (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ) ≤ 1 - ε
accumulation :
  Tendsto (β := ℝ≥0) (fun (n : ℕ) ↦ ((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i) : ℝ≥0)) atTop
    (𝓝 ((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), U i)) : Filter ℝ≥0)
⊢ Tendsto (fun (a : ℕ) ↦ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ a), U i)) : ℝ)) atTop (𝓝 (↑1 : ℝ))

Tactic:
rw [Cov, coeFn_univ, ← NNReal.tendsto_coe] at accumulation

Premises used: NNReal.tendsto_coe, MeasureTheory.ProbabilityMeasure.coeFn_univ
-- 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)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), U i) univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : (k : ℕ) → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction :
  ∀ (k : ℕ), (↑((μ k : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), U i)) : ℝ≥0∞) ≤ 1 - (↑(NNReal.mk ε hε') : ℝ≥0∞)
μlim : ProbabilityMeasure 𝓧
left : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
Measurebound : ∀ (n : ℕ), (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ) ≤ 1 - ε
accumulation :
  Tendsto (fun (a : ℕ) ↦ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ a), U i)) : ℝ)) atTop (𝓝 (↑1 : ℝ))
simp : 0 < ε
⊢ 1 - ε / 2 < 1

Tactic:
simp only [sub_lt_self_iff, Nat.ofNat_pos, div_pos_iff_of_pos_right]

Premises used: div_pos_iff_of_pos_right, Nat.ofNat_pos, sub_lt_self_iff
-- 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)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), U i) univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : (k : ℕ) → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction :
  ∀ (k : ℕ), (↑((μ k : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), U i)) : ℝ≥0∞) ≤ 1 - (↑(NNReal.mk ε hε') : ℝ≥0∞)
μlim : ProbabilityMeasure 𝓧
left : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
Measurebound : ∀ (n : ℕ), (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ) ≤ 1 - ε
accumulation :
  Tendsto (fun (a : ℕ) ↦ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ a), U i)) : ℝ)) atTop (𝓝 (↑1 : ℝ))
⊢ 0 < ε

Tactic:
positivity
-- 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)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), U i) univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : (k : ℕ) → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction :
  ∀ (k : ℕ), (↑((μ k : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), U i)) : ℝ≥0∞) ≤ 1 - (↑(NNReal.mk ε hε') : ℝ≥0∞)
μlim : ProbabilityMeasure 𝓧
left : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
Measurebound : ∀ (n : ℕ), (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ) ≤ 1 - ε
accumulation :
  Tendsto (fun (a : ℕ) ↦ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ a), U i)) : ℝ)) atTop (𝓝 (↑1 : ℝ))
⊢ ∀ᶠ (n : ℕ) in atTop, 1 - ε / 2 ≤ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ)

Tactic:
have exceeds_bound : ∀ᶠ n in atTop, (1 - ε / 2 : ℝ) ≤ μlim (⋃ i ≤ n, U i) :=
      Tendsto.eventually_const_le (v := 1)
        (by simp only [sub_lt_self_iff, Nat.ofNat_pos, div_pos_iff_of_pos_right]; positivity)
        accumulation
-- 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)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), U i) univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : (k : ℕ) → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction :
  ∀ (k : ℕ), (↑((μ k : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), U i)) : ℝ≥0∞) ≤ 1 - (↑(NNReal.mk ε hε') : ℝ≥0∞)
μlim : ProbabilityMeasure 𝓧
left : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
Measurebound : ∀ (n : ℕ), (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ) ≤ 1 - ε
accumulation :
  Tendsto (fun (a : ℕ) ↦ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ a), U i)) : ℝ)) atTop (𝓝 (↑1 : ℝ))
exceeds_bound : ∀ᶠ (n : ℕ) in atTop, 1 - ε / 2 ≤ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ)
suffices : ∀ᶠ (n : ℕ) in atTop, False
⊢ False

Tactic:
suffices ∀ᶠ n : ℕ in atTop, False from this.exists.choose_spec
-- 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)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), U i) univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : (k : ℕ) → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction :
  ∀ (k : ℕ), (↑((μ k : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), U i)) : ℝ≥0∞) ≤ 1 - (↑(NNReal.mk ε hε') : ℝ≥0∞)
μlim : ProbabilityMeasure 𝓧
left : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
Measurebound : ∀ (n : ℕ), (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ) ≤ 1 - ε
accumulation :
  Tendsto (fun (a : ℕ) ↦ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ a), U i)) : ℝ)) atTop (𝓝 (↑1 : ℝ))
exceeds_bound : ∀ᶠ (n : ℕ) in atTop, 1 - ε / 2 ≤ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ)
h : ∀ (n : ℕ) (hn : 1 - ε / 2 ≤ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ)), False
⊢ ∀ᶠ (n : ℕ) in atTop, False

Tactic:
filter_upwards [exceeds_bound] with n hn
-- 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)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), U i) univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : (k : ℕ) → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction :
  ∀ (k : ℕ), (↑((μ k : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ k), U i)) : ℝ≥0∞) ≤ 1 - (↑(NNReal.mk ε hε') : ℝ≥0∞)
μlim : ProbabilityMeasure 𝓧
left : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
Measurebound : ∀ (n : ℕ), (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ) ≤ 1 - ε
accumulation :
  Tendsto (fun (a : ℕ) ↦ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ a), U i)) : ℝ)) atTop (𝓝 (↑1 : ℝ))
exceeds_bound : ∀ᶠ (n : ℕ) in atTop, 1 - ε / 2 ≤ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ)
n : ℕ
hn : 1 - ε / 2 ≤ (↑((μlim : (a : Set 𝓧) → ℝ≥0) (⋃ (i : ℕ), ⋃ (_ : i ≤ n), U i)) : ℝ)
linarith : 2 / 2 = 1
linarith₁ : HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : GT.gt (α := ℝ) 2 0
linarith₃ : 2 / 2 = 1
linarith₄ : HMul.hMul (α := ℝ) 1 2 = 2
linarith₅ : GT.gt (α := ℝ) 2 0
linarith₆ : 0 - ε + (HMul.hMul (α := ℝ) 2 1 - 1 * ε - (HMul.hMul (α := ℝ) 2 1 - 2 * ε)) = 0
⊢ False

Tactic:
linarith [hn.trans <| Measurebound n]

Premises used: LE.le.trans
-- 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)
rw :
  ∀ (ε : ℝ≥0∞),
    0 < 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
⊢ IsTightMeasureSet.{u_1} (𝓧 := 𝓧) (m𝓧 := m𝓧) {x : Measure 𝓧 | ∃ μ ∈ S, (↑μ : Measure 𝓧) = x}

Tactic:
rw [isTightMeasureSet_iff_exists_isCompact_measure_compl_le]

Premises used: MeasureTheory.isTightMeasureSet_iff_exists_isCompact_measure_compl_le
-- 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)
inl :
  ∀ (hempty : IsEmpty 𝓧) (ε : ℝ≥0∞),
    0 < 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
inr :
  ∀ (hnonempty : Nonempty 𝓧) (ε : ℝ≥0∞),
    0 < 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
⊢ ∀ (ε : ℝ≥0∞),
    0 < 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 isEmpty_or_nonempty 𝓧 with hempty | hnonempty

Premises used: isEmpty_or_nonempty
-- 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)
hempty : IsEmpty 𝓧
⊢ Eq.{u_1 + 1} (α := Set 𝓧) univ ∅

Tactic:
rw [← univ_eq_empty_iff] at hempty

Premises used: Set.univ_eq_empty_iff
-- 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)
hempty : Eq.{u_1 + 1} (α := Set 𝓧) univ ∅
ε : ℝ≥0∞
εpos : 0 < ε
⊢ ∀ (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∞) ∅ᶜ ≤ ε

Tactic:
simp [hempty]
-- 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)
hempty : Eq.{u_1 + 1} (α := Set 𝓧) univ ∅
⊢ ∀ (ε : ℝ≥0∞),
    0 < 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:
exact fun ε εpos ↦ ⟨∅, isCompact_empty, by simp [hempty]⟩
-- 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
inr :
  ∀ (D : ℕ → 𝓧) (hD : DenseRange D) (ε : ℝ≥0∞),
    0 < 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
⊢ ∀ (ε : ℝ≥0∞),
    0 < 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:
obtain ⟨D, hD⟩ := exists_dense_seq 𝓧

Premises used: TopologicalSpace.exists_dense_seq
-- 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), StrictAnti u ∧ (∀ (n : ℕ), 0 < u n) ∧ Tendsto u atTop (𝓝 0)

Tactic:
exists_seq_strictAnti_tendsto (0 : ℝ)

Premises used: exists_seq_strictAnti_tendsto, 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
inr :
  ∀ (u : ℕ → ℝ) (hu_anti : StrictAnti u) (hu_pos : ∀ (n : ℕ), 0 < u n) (hu : Tendsto u atTop (𝓝 0)) (ε : ℝ≥0∞),
    0 < 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
⊢ ∀ (ε : ℝ≥0∞),
    0 < 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:
obtain ⟨u, hu_anti, hu_pos, hu⟩ : ∃ u, StrictAnti u ∧ (∀ n, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ)

Premises used: exists_seq_strictAnti_tendsto, 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)
m : ℕ
rw :
  ∀ (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),
    Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ
⊢ Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ

Tactic:
rw [denseRange_iff] at hD

Premises used: Metric.denseRange_iff
-- 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
h : ∀ (p : 𝓧), Membership.mem (γ := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) p ↔ Membership.mem (γ := Set 𝓧) univ p
⊢ Eq.{u_1 + 1} (α := Set 𝓧) (⋃ (i : ℕ), ball (D i) (u m)) univ

Tactic:
ext p