mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.Intersectivity

Goal state before:
pp: case refine_3
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n ↦ (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 ↦ f x_1 x) atTop ∂μ ≤ μ univ
x : α
hxN : x ∉ N
hx : r / μ univ ≤ limsup (fun x_1 ↦ f x_1 x) atTop
⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)

Tactic:
refine ⟨{n | x ∈ s n}, fun hxs ↦ ?_, fun u hux hu ↦ ?_⟩

Goal state after:
case refine_3.refine_1
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n ↦ (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 ↦ f x_1 x) atTop ∂μ ≤ μ univ
x : α
hxN : x ∉ N
hx : r / μ univ ≤ limsup (fun x_1 ↦ f x_1 x) atTop
hxs : {n | x ∈ s n}.Finite
⊢ False

case refine_3.refine_2
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n ↦ (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 ↦ f x_1 x) atTop ∂μ ≤ μ univ
x : α
hxN : x ∉ N
hx : r / μ univ ≤ limsup (fun x_1 ↦ f x_1 x) atTop
u : Set ℕ
hux : u ⊆ {n | x ∈ s n}
hu : u.Finite
⊢ 0 < μ (⋂ n ∈ u, s n)
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_top : p ≠ ∞
n : ℕ
f : Fin n → α → β
hf : ∀ (i : Fin n), MemLp (f i) p μ
⊢ UnifTight f p μ

Tactic:
revert f
  induction n with
  | zero => exact fun {f} hf ↦ unifTight_of_subsingleton hp_top hf
  | succ n h =>
    intro f hfLp ε hε
    by_cases hε_top : ε = ∞
    · exact ⟨∅, (by simp), fun _ => hε_top.symm ▸ le_top⟩
    let g : Fin n → α → β := fun k => f k.castSucc
    have hgLp : ∀ i, MemLp (g i) p μ := fun i => hfLp i.castSucc
    obtain ⟨S, hμS, hFε⟩ := h hgLp hε
    obtain ⟨s, _, hμs, hfε⟩ :=
      (hfLp (Fin.last n)).exists_eLpNorm_indicator_compl_lt hp_top (coe_ne_zero.2 hε.ne')
    refine ⟨s ∪ S, (by finiteness), fun i => ?_⟩
    by_cases! hi : i.val < n
    · rw [show f i = g ⟨i.val, hi⟩ from rfl, compl_union, ← indicator_indicator]
      apply (eLpNorm_indicator_le _).trans
      exact hFε (Fin.castLT i hi)
    · obtain rfl : i = Fin.last n := Fin.ext (le_antisymm i.is_le hi)
      rw [compl_union, inter_comm, ← indicator_indicator]
      exact (eLpNorm_indicator_le _).trans hfε.le

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ) →ₚ[ℝ] ℝ
a : ℝ≥0
f : α →C_c ℝ≥0
⊢ NNReal.mk (Λ (toRealLinearMap (a • f))) ⋯ = (RingHom.id ℝ≥0) a • NNReal.mk (Λ (toRealLinearMap f)) ⋯

Tactic:
simp [NNReal.smul_def]; rfl

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ (∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p)) * (volume.restrict (Ioo (-π) π)).real univ • 1 =
    2 * π * ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p)

Tactic:
measureReal_restrict_apply MeasurableSet.univ,

Goal state after:
p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ (∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p)) * volume.real (univ ∩ Ioo (-π) π) • 1 =
    2 * π * ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p)
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ (∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p)) * (volume.restrict (Ioo (-π) π)).real univ • 1 =
    2 * π * ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p)

Tactic:
measureReal_restrict_apply MeasurableSet.univ,

Goal state after:
p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ (∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p)) * volume.real (univ ∩ Ioo (-π) π) • 1 =
    2 * π * ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := ⋯
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
x : X
hx : x ∈ s₀
this : s₀ ⊆ (tsupport ⇑(f 0))ᶜ
⊢ x ∈ (support ⇑(f 0))ᶜ

Tactic:
exact mem_of_subset_of_mem (subset_trans this (compl_subset_compl_of_subset subset_closure))
        hx

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := ⋯
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
x : X
hx : x ∈ s₁
this : s₁ ⊆ (tsupport ⇑(f 1))ᶜ
⊢ x ∈ (support ⇑(f 1))ᶜ

Tactic:
exact mem_of_subset_of_mem (subset_trans this (compl_subset_compl_of_subset subset_closure))
        hx

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
b : ℝ≥0∞
hb : ∀ (i : Compacts X), b ≤ ⨅ (_ : ↑K ⊆ interior ↑i), ↑(rieszContentAux Λ i)
this : b < ∞
ε : ℝ≥0
hε : 0 < ↑ε
f : X →C_c ℝ≥0
hfleoneonK : ∀ x ∈ K, 1 ≤ f x
hfle : ↑(Λ f) < ↑(sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x}) + NNReal.mk ↑ε ⋯)
α : ℝ≥0
hα : 1 < α
K' : Set X := ⇑f ⁻¹' Ici α⁻¹
hKK' : ↑K ⊆ interior K'
hK'cp : IsCompact K'
hb' : b ≤
  ⨅ (_ : ↑K ⊆ interior ↑{ carrier := K', isCompact' := hK'cp }),
    ↑(rieszContentAux Λ { carrier := K', isCompact' := hK'cp }) :=
  hb { carrier := K', isCompact' := hK'cp }
⊢ b.toNNReal ≤ Λ (α • f)

Tactic:
simp only [Compacts.coe_mk, le_iInf_iff] at hb'

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
b : ℝ≥0∞
hb : ∀ (i : Compacts X), b ≤ ⨅ (_ : ↑K ⊆ interior ↑i), ↑(rieszContentAux Λ i)
this : b < ∞
ε : ℝ≥0
hε : 0 < ↑ε
f : X →C_c ℝ≥0
hfleoneonK : ∀ x ∈ K, 1 ≤ f x
hfle : ↑(Λ f) < ↑(sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x}) + NNReal.mk ↑ε ⋯)
α : ℝ≥0
hα : 1 < α
K' : Set X := ⇑f ⁻¹' Ici α⁻¹
hKK' : ↑K ⊆ interior K'
hK'cp : IsCompact K'
hb' : ↑K ⊆ interior K' → b ≤ ↑(rieszContentAux Λ { carrier := K', isCompact' := hK'cp })
⊢ b.toNNReal ≤ Λ (α • f)
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
⊢ RegularSpace X

Tactic:
by
  rw [regularSpace_iff]
  intro s a hs ha
  obtain ⟨f, cf, hf, hhf⟩ := CompletelyRegularSpace.completely_regular a s hs ha
  apply disjoint_of_map (f := f)
  apply Disjoint.mono (cf.tendsto_nhdsSet_nhds hhf) cf.continuousAt
  exact disjoint_nhds_nhds.mpr (hf.symm ▸ zero_ne_one).symm

Goal state after:
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : T35Space X
a b : X
hab : stoneCechUnit a = stoneCechUnit b
⊢ a = b

Tactic:
contrapose hab

Goal state after:
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : T35Space X
a b : X
hab : ¬a = b
⊢ ¬stoneCechUnit a = stoneCechUnit b
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case neg
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ μ B ≤
      κ (thickening (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal B) +
          (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ) +
        ε ∧
    κ B ≤
      μ (thickening (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal B) +
          (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ) +
        ε

Tactic:
← hs_add_r',

Goal state after:
case neg
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ μ B ≤ κ (thickening (s.toReal + r.toReal) B) + (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ) + ε ∧
    κ B ≤ μ (thickening (s.toReal + r.toReal) B) + (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ) + ε
Module: Mathlib.MeasureTheory.Measure.FiniteMeasurePi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
s : (i : ι) → Set (α i)
smeas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (i : ι), (μ i) (frontier (s i)) = 0
s' : (i : ι) → Set (α i)
s'meas : ∀ (i : ι), MeasurableSet (s' i)
hs' : ∀ (i : ι), (μ i) (frontier (s' i)) = 0
i : ι
⊢ (μ i) (frontier ((fun i ↦ s i ∩ s' i) i)) = 0

Tactic:
simp_rw [null_iff_toMeasure_null] at hs hs' ⊢

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
s : (i : ι) → Set (α i)
smeas : ∀ (i : ι), MeasurableSet (s i)
s' : (i : ι) → Set (α i)
s'meas : ∀ (i : ι), MeasurableSet (s' i)
i : ι
hs : ∀ (i : ι), ↑(μ i) (frontier (s i)) = 0
hs' : ∀ (i : ι), ↑(μ i) (frontier (s' i)) = 0
⊢ ↑(μ i) (frontier (s i ∩ s' i)) = 0
Module: Mathlib.MeasureTheory.Measure.FiniteMeasurePi

Goal state before:
pp: case h
ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) := ⋯
this : IsPiSystem S
⊢ ∀ (u : Set ((i : ι) → α i)), IsOpen[Pi.topologicalSpace] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u

Tactic:
let : ∀ i, PseudoMetricSpace (α i) :=
      fun i ↦ TopologicalSpace.pseudoMetrizableSpacePseudoMetric (α i)

Goal state after:
case h
ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
this✝ : IsPiSystem S
this : (i : ι) → PseudoMetricSpace (α i) := fun i ↦ pseudoMetrizableSpacePseudoMetric (α i)
⊢ ∀ (u : Set ((i : ι) → α i)), IsOpen[Pi.topologicalSpace] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case a
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν : Measure Ω
hLP : levyProkhorovEDist μ ν = 0
s : Set Ω
s_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s
hμs : ∃ δ > 0, μ (thickening δ s) ≠ ∞
hνs : ∃ δ > 0, ν (thickening δ s) ≠ ∞
⊢ ν s ≤ μ s

Tactic:
exact measure_le_measure_closure_of_levyProkhorovEDist_eq_zero
      (levyProkhorovEDist_comm μ ν ▸ hLP) s_closed.measurableSet hμs |>.trans <|
      le_of_eq (congr_arg _ s_closed.closure_eq)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
h : ∀ (n : Fin N), (y n - ε') * μ.real (E n) ≤ ∫ (x : X) in E n, f x ∂μ
⊢ ∀ (i : Fin N), IntegrableOn (⇑f) (E i) (rieszContent (toNNRealLinear Λ)).measure

Tactic:
exact fun _ ↦
          Integrable.integrableOn <| Continuous.integrable_of_hasCompactSupport f.1.2 f.2

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: α : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
f : α →ᵇ ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] ⇑f
hf : HasFiniteIntegral (⇑f) μ
⊢ ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioc 0 ‖f‖, μ.real {a | t ≤ f a}

Tactic:
by
  rw [Integrable.integral_eq_integral_Ioc_meas_le (M := ‖f‖) ?_ f_nn ?_]
  · exact ⟨f.continuous.measurable.aestronglyMeasurable, hf⟩
  · exact Eventually.of_forall (fun x ↦ BoundedContinuousFunction.apply_le_norm f x)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toMeasure
Ps : ℕ → ProbabilityMeasure Ω := toMeasure ∘ μs
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
εs : ℕ → ℝ
left✝ : StrictAnti εs
εs_pos : ∀ (n : ℕ), 0 < εs n
εs_lim : Tendsto εs atTop (𝓝 0)
ε_of_room : Tendsto (fun x ↦ dist (μs x) ν + εs x) atTop (𝓝 (0 + 0))
n : ℕ
⊢ 0 < dist (μs n) ν + εs n

Tactic:
linarith [εs_pos n, dist_nonneg (x := μs n) (y := ν)]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case inr.refine_2
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
ε : ℝ
ε_pos : 0 < ε
h✝ : Nonempty Ω
xs : ℕ → Ω
xs_dense : DenseRange xs
half_ε_pos : 0 < ε / 2
Bs : ℕ → Set Ω := fun n ↦ ball (xs n) (ε / 2)
As : ℕ → Set Ω := disjointed Bs
⊢ ∀ (n : ℕ), Bornology.IsBounded (As n)

Tactic:
exact fun n ↦ Bornology.IsBounded.subset isBounded_ball <| disjointed_subset Bs n

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
exhaust : ∀ (U : Set ℝ≥0∞), 0 ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → ↑P (⋃ i, ⋃ (_ : i ≥ n), Es i) ∈ U
⊢ ∃ N, ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)

Tactic:
obtain ⟨N, hN⟩ := exhaust (Iio (ENNReal.ofReal (ε / 3))) third_ε_pos' isOpen_Iio

Goal state after:
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
exhaust : ∀ (U : Set ℝ≥0∞), 0 ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → ↑P (⋃ i, ⋃ (_ : i ≥ n), Es i) ∈ U
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → ↑P (⋃ i, ⋃ (_ : i ≥ n), Es i) ∈ Iio (ENNReal.ofReal (ε / 3))
⊢ ∃ N, ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e_a
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2 = 2⁻¹ * r * 2 * (2 * r⁻¹)

pp: case e_a
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ ∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ = 1 - ∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ

Tactic:
· ring_nf
      rw [mul_inv_cancel₀ (by positivity), one_mul]

Goal state after:
case e_a
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ ∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ = 1 - ∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case refine_1
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
Λ' : (E →C_c ℝ) →ₚ[ℝ] ℝ := { toFun := Λ, map_add' := ⋯, map_smul' := ⋯, monotone' := ⋯ }
μlim : Measure E := RealRMK.rieszMeasure Λ'
μlim_le : μlim univ ≤ ENNReal.ofReal ↑C
μlim' : FiniteMeasure E := ⟨μlim, ⋯⟩
⊢ (μlim univ).toNNReal ≤ C

Tactic:
rw [show C = (ENNReal.ofReal ↑C).toNNReal by simp]

Goal state after:
case refine_1
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
Λ' : (E →C_c ℝ) →ₚ[ℝ] ℝ := { toFun := Λ, map_add' := ⋯, map_smul' := ⋯, monotone' := ⋯ }
μlim : Measure E := RealRMK.rieszMeasure Λ'
μlim_le : μlim univ ≤ ENNReal.ofReal ↑C
μlim' : FiniteMeasure E := ⟨μlim, ⋯⟩
⊢ (μlim univ).toNNReal ≤ (ENNReal.ofReal ↑C).toNNReal
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case refine_1
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
h_limsup_le :
  ∀ (r : ℝ),
    0 < r →
      limsup (fun n ↦ (μ n).real {x | r < |⟪z, x⟫|}) atTop ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)‖
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
⊢ ∃ N, ∀ (n : ℝ), N ≤ n → 2⁻¹ * |n| * ‖∫ (t : ℝ) in -(2 * n⁻¹)..2 * n⁻¹, 1 - f (t • z)‖ < ε

Tactic:
refine ⟨4 * δ⁻¹, fun r hrδ ↦ ?_⟩

Goal state after:
case refine_1
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
h_limsup_le :
  ∀ (r : ℝ),
    0 < r →
      limsup (fun n ↦ (μ n).real {x | r < |⟪z, x⟫|}) atTop ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)‖
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ r
⊢ 2⁻¹ * |r| * ‖∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - f (t • z)‖ < ε