mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : NormalSpace X
inst✝ : R0Space X
x : X
K : Set X
hK : IsClosed[inst✝²] K
hx : x ∉ K
cx : IsClosed[inst✝²] (closure[inst✝²] {x})
d : Disjoint (closure[inst✝²] {x}) K
f : X → ℝ
cf : Continuous[inst✝², _] f
hfx : EqOn (⇑{ toFun := f, continuous_toFun := cf }) 0 (closure[inst✝²] {x})
hfK : EqOn (⇑{ toFun := f, continuous_toFun := cf }) 1 K
hficc : ∀ (x : X), { toFun := f, continuous_toFun := cf } x ∈ Icc 0 1
g : X → ↑I := fun x ↦ ⟨f x, ⋯⟩
cg : Continuous[inst✝², _] g
⊢ ∃ f, Continuous[inst✝², _] f ∧ f x = 0 ∧ EqOn f 1 K

Tactic:
have hgx : g x = 0 := Subtype.ext (hfx (subset_closure (mem_singleton x)))

Goal state after:
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : NormalSpace X
inst✝ : R0Space X
x : X
K : Set X
hK : IsClosed[inst✝²] K
hx : x ∉ K
cx : IsClosed[inst✝²] (closure[inst✝²] {x})
d : Disjoint (closure[inst✝²] {x}) K
f : X → ℝ
cf : Continuous[inst✝², _] f
hfx : EqOn (⇑{ toFun := f, continuous_toFun := cf }) 0 (closure[inst✝²] {x})
hfK : EqOn (⇑{ toFun := f, continuous_toFun := cf }) 1 K
hficc : ∀ (x : X), { toFun := f, continuous_toFun := cf } x ∈ Icc 0 1
g : X → ↑I := fun x ↦ ⟨f x, ⋯⟩
cg : Continuous[inst✝², _] g
hgx : g x = 0
⊢ ∃ f, Continuous[inst✝², _] f ∧ f x = 0 ∧ EqOn f 1 K
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝² : TopologicalSpace X
Y : Type v
inst✝¹ : TopologicalSpace Y
inst✝ : CompletelyRegularSpace Y
f : X → Y
hf : IsInducing f
x : X
K : Set Y
hK : IsClosed[inst✝¹] K
hxK : f x ∉ K
g : Y → ↑I
hcf : Continuous[inst✝¹, _] g
egfx : g (f x) = 0
hgK : EqOn g 1 K
| EqOn (g ∘ f) 1 (f ⁻¹' K)

Tactic:
conv =>

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

Goal state before:
pp: ι : Type u_1
X : Type u_2
t : ι → TopologicalSpace X
ht : ∀ (i : ι), CompletelyRegularSpace X
this : TopologicalSpace X := ⨅ i, t i
x : X
I' : Finset ι
V U : ↥I' → Set X
hUV : ∀ (i : ↥I'), U i ⊆ V i
fs : ↥I' → X → ↑I
hfs : ∀ (i : ↥I'), Continuous[t ↑i, _] (fs i)
hxfs : ∀ (i : ↥I'), fs i x = 0
hfsU : ∀ (i : ↥I') ⦃x : X⦄, x ∈ (U i)ᶜ → fs i x = ⊤
| ∀ ⦃x : X⦄, x ∈ (⋂ i, V i)ᶜ → I'.attach.sup fs x = ⊤

Tactic:
conv =>

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

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
f : X → ↑I
hf : Continuous[inst✝¹, _] f
efx : f x = 0
hfU : EqOn f 1 Uᶜ
| EqOn f 1 Uᶜ

Tactic:
conv =>

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
⊢ dist (μs n) ν + εs n ∈ Ioi 0

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

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar

Goal state before:
pp: G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
φ : G ≃ₜ* G
smul : haar = haar.haarScalarFactor μ • μ
| haar.haarScalarFactor (map (⇑φ) haar) = μ.haarScalarFactor (map (⇑φ) μ)

Tactic:
conv =>

Goal state after:
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
φ : G ≃ₜ* G
smul : haar = haar.haarScalarFactor μ • μ
| haar.haarScalarFactor μ • μ
Module: Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar

Goal state before:
pp: G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
φ : G ≃ₜ* G
smul : haar = haar.haarScalarFactor μ • μ
| (haar.haarScalarFactor μ • μ).haarScalarFactor (map (⇑φ) haar) = μ.haarScalarFactor (map (⇑φ) μ)

Tactic:
conv =>

Goal state after:
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
φ : G ≃ₜ* G
smul : haar = haar.haarScalarFactor μ • μ
| haar.haarScalarFactor μ • μ
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case h
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
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
x : E
⊢ (fun x ↦ -‖g.toBoundedContinuousFunction‖) x ≤ (fun x ↦ g x) x

Tactic:
apply neg_le_of_abs_le

Goal state after:
case h
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
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
x : E
⊢ |(fun x ↦ g x) x| ≤ ‖g.toBoundedContinuousFunction‖
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
k : ℕ
hk : finrank ℝ E = 2 * k + 1
x : E
r : ℝ
⊢ volume (ball x r) = ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (π ^ k * 2 ^ (k + 1) / ↑(finrank ℝ E)‼)

Tactic:
by
  have : Nontrivial E := Module.nontrivial_of_finrank_pos (R := ℝ) (hk ▸ (2 * k).succ_pos)
  rw [volume_ball, hk, pow_succ (√π), pow_mul, sq_sqrt pi_nonneg, mul_div_assoc, mul_div_assoc]
  congr 3
  suffices √π / (↑(2 * k + 1)‼ * √π / 2 ^ (k + 1)) = 2 ^ (k + 1) / ↑(2 * k + 1)‼ by
    simpa [add_div, add_right_comm, -one_div, Gamma_nat_add_one_add_half]
  field

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

Goal state before:
pp: case neg
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
n : ℕ
hn : ¬n = 0
⊢ ∑ i ∈ Finset.range n, preVariationFun f (s i) ≤ preVariationFun f (⋃ i, s i)

Tactic:
refine ENNReal.le_of_forall_pos_le_add fun ε' hε' hsnetop ↦ ?_

Goal state after:
case neg
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
n : ℕ
hn : ¬n = 0
ε' : ℝ≥0
hε' : 0 < ε'
hsnetop : preVariationFun f (⋃ i, s i) < ∞
⊢ ∑ i ∈ Finset.range n, preVariationFun f (s i) ≤ preVariationFun f (⋃ i, s i) + ↑ε'
Module: Mathlib.MeasureTheory.Measure.Support

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : MeasurableSpace X
⊢ Measure.support 0 = ∅

Tactic:
simp [Measure.support]

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

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
μ : Measure X
inst✝ : OpensMeasurableSpace X
s : Set X
x : X
⊢ x ∈ (μ.restrict s).support ↔ ∃ᶠ (u : Set X) in (𝓝[s] x).smallSets, 0 < μ u

Tactic:
by
  rw [nhds_basis_opens x |>.mem_measureSupport,
    (nhdsWithin_basis_open x s).frequently_smallSets pos_mono]
  grind [IsOpen.measurableSet, restrict_apply]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : HereditarilyLindelofSpace 𝕜
inst✝⁴ : CompleteSpace 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : BorelSpace 𝕜
inst✝¹ : NormedAlgebra 𝕜 ℂ
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
⊢ DifferentiableOn ℂ (resolventTransform μ) (⇑(algebraMap 𝕜 ℂ) '' μ.support)ᶜ

Tactic:
intro z hz
    exact (hasDerivAt_resolventTransform z hz).differentiableAt.differentiableWithinAt

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

Goal state before:
pp: X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (f n ∆ t) < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n ↦ (μ (accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), μ.real (f n ∆ g n) < ε / (2 * (↑N + 1))
⊢ (↑N + 1) * (ε / 2) * (1 / (↑N + 1)) ≤ ε / 2

Tactic:
mul_comm ((N : ℝ) + 1),

Goal state after:
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (f n ∆ t) < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n ↦ (μ (accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), μ.real (f n ∆ g n) < ε / (2 * (↑N + 1))
⊢ ε / 2 * (↑N + 1) * (1 / (↑N + 1)) ≤ ε / 2
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
⊢ IsCompact {μ | ∀ (n : ℕ), μ (K n)ᶜ ≤ u n}

Tactic:
apply (ProbabilityMeasure.toFiniteMeasure_isEmbedding E).isCompact_iff.2
  have : ProbabilityMeasure.toFiniteMeasure '' {μ | ∀ (n : ℕ), μ (K n)ᶜ ≤ u n}
      = {μ : FiniteMeasure E | μ.mass = 1 ∧ ∀ n, μ (K n)ᶜ ≤ u n} := by
    ext μ
    simp only [mem_image, mem_ofPred_eq]
    refine ⟨?_, ?_⟩
    · rintro ⟨ν, hν, rfl⟩
      simpa using! hν
    · rintro ⟨hμ, h'μ⟩
      let ν : ProbabilityMeasure E := ⟨μ, isProbabilityMeasure_iff_real.2 (by simpa using! hμ)⟩
      have : ν.toFiniteMeasure = μ := by ext; rfl
      exact ⟨ν, by simpa [← this] using! h'μ , this⟩
  rw [this]
  exact isCompact_setOfPred_finiteMeasure_mass_eq_compl_isCompact_le 1 hu hK h

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

Goal state before:
pp: X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
this : MeasurableSet s ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (s ∆ t) < ε
ε : ℝ
ε_pos : 0 < ε
t : Set X
t_mem : t ∈ 𝒜
hμst : μ.real (s ∆ t) < ε
⊢ ∃ t ∈ 𝒜, μ (s ∆ t) < ENNReal.ofReal ε

Tactic:
exact ⟨t, t_mem, (lt_ofReal_iff_toReal_lt (measure_ne_top _ _)).2 hμst⟩

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

Goal state before:
pp: X : Type u_1
m : MeasurableSpace X
μ : Measure X
inst✝¹ : CountablyGenerated X
inst✝ : SigmaFinite μ
h : (countableGeneratingSet X).Countable
hgen : MeasurableSpace.generateFrom (countableGeneratingSet X) = m
⊢ ∃ 𝒜, 𝒜.Countable ∧ μ.MeasureDense 𝒜

Tactic:
let 𝒜 := (countableGeneratingSet X) ∪ {μ.toFiniteSpanningSetsIn.set n | n : ℕ}

Goal state after:
X : Type u_1
m : MeasurableSpace X
μ : Measure X
inst✝¹ : CountablyGenerated X
inst✝ : SigmaFinite μ
h : (countableGeneratingSet X).Countable
hgen : MeasurableSpace.generateFrom (countableGeneratingSet X) = m
𝒜 : Set (Set X) := countableGeneratingSet X ∪ {x | ∃ n, μ.toFiniteSpanningSetsIn.set n = x}
⊢ ∃ 𝒜, 𝒜.Countable ∧ μ.MeasureDense 𝒜
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ i, f i
A : ℕ → Finset (Set α) := fun i ↦ hC.disjointOfDiff ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ ↑(A i)
⊢ (OuterMeasure.ofFunction ⇑m ⋯) (t ∩ s) + (OuterMeasure.ofFunction ⇑m ⋯) (t \ s) ≤ ∑' (i : ℕ), m (f i)

Tactic:
have h_m_eq i : m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u :=
    eq_add_disjointOfDiff_of_subset hC (hC.inter_mem (f i) (hf i) s hs) (hf i) inter_subset_left

Goal state after:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ i, f i
A : ℕ → Finset (Set α) := fun i ↦ hC.disjointOfDiff ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ ↑(A i)
h_m_eq : ∀ (i : ℕ), m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u
⊢ (OuterMeasure.ofFunction ⇑m ⋯) (t ∩ s) + (OuterMeasure.ofFunction ⇑m ⋯) (t \ s) ≤ ∑' (i : ℕ), m (f i)
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : CountablyGenerated X
inst✝ : SFinite μ
this✝ : IsSeparable (μ.restrict μ.sigmaFiniteSet)
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : (μ.restrict μ.sigmaFiniteSet).MeasureDense 𝒜
ℬ : Set (Set X) := {x | ∃ s ∈ 𝒜, s ∩ μ.sigmaFiniteSet = x}
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
hs : (μ.restrict μ.sigmaFiniteSetᶜ) s = 0
this : (μ.restrict μ.sigmaFiniteSet) s ≠ ∞
t : Set X
t_mem : t ∈ 𝒜
ht : (μ.restrict μ.sigmaFiniteSet) (s ∆ t) < ENNReal.ofReal ε
⊢ μ (μ.sigmaFiniteSetᶜ ∩ s ∆ (t ∩ μ.sigmaFiniteSet)) = 0

Tactic:
inter_symmDiff_distrib_left,

Goal state after:
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : CountablyGenerated X
inst✝ : SFinite μ
this✝ : IsSeparable (μ.restrict μ.sigmaFiniteSet)
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : (μ.restrict μ.sigmaFiniteSet).MeasureDense 𝒜
ℬ : Set (Set X) := {x | ∃ s ∈ 𝒜, s ∩ μ.sigmaFiniteSet = x}
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
hs : (μ.restrict μ.sigmaFiniteSetᶜ) s = 0
this : (μ.restrict μ.sigmaFiniteSet) s ≠ ∞
t : Set X
t_mem : t ∈ 𝒜
ht : (μ.restrict μ.sigmaFiniteSet) (s ∆ t) < ENNReal.ofReal ε
⊢ μ ((μ.sigmaFiniteSetᶜ ∩ s) ∆ (μ.sigmaFiniteSetᶜ ∩ (t ∩ μ.sigmaFiniteSet))) = 0
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
| (∫ (x : X), f x ∂μ).ofLp.1 = ∫ (x : X), (f x).fst ∂μ

Tactic:
conv =>

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
| (WithLp.prodContinuousLinearEquiv q ℝ E F) (∫ (x : X), f x ∂μ)
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
| (∫ (x : X), f x ∂μ).ofLp.2 = ∫ (x : X), (f x).snd ∂μ

Tactic:
conv =>

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
| (WithLp.prodContinuousLinearEquiv q ℝ E F) (∫ (x : X), f x ∂μ)
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case refine_2.refine_1
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
a : E
s : Set X
ms : MeasurableSet s
hμs✝ : μ s < ∞
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
μs_pow_nonneg : 0 ≤ μ.real s ^ (1 / p.toReal)
approx_a_pos : 0 < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
b : E
b_mem : b ∈ u
hb : ‖a - b‖ < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
⊢ ∃ b ∈ D, ‖↑(simpleFunc.indicatorConst p ms ⋯ a) - b‖ < ε

Tactic:
rcases SeminormedAddCommGroup.mem_closure_iff.1
        (h𝒜₀.indicatorConstLp_subset_closure p b ⟨s, ms, hμs, rfl⟩)
          (ε / 3) (by linarith [ε_pos]) with ⟨-, ⟨t, ht, hμt, rfl⟩, hst⟩

Goal state after:
case refine_2.refine_1
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
a : E
s : Set X
ms : MeasurableSet s
hμs✝ : μ s < ∞
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
μs_pow_nonneg : 0 ≤ μ.real s ^ (1 / p.toReal)
approx_a_pos : 0 < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
b : E
b_mem : b ∈ u
hb : ‖a - b‖ < ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))
t : Set X
ht : t ∈ 𝒜₀
hμt : μ t ≠ ∞
hst : ‖indicatorConstLp p ms hμs b - indicatorConstLp p ⋯ hμt b‖ < ε / 3
⊢ ∃ b ∈ D, ‖↑(simpleFunc.indicatorConst p ms ⋯ a) - b‖ < ε
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝³ : PseudoMetricSpace 𝓧
inst✝² : OpensMeasurableSpace 𝓧
inst✝¹ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst✝ : 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), ⋃ i, ball (D i) (u m) = univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
km : ℕ → ℕ
bigK : Set 𝓧 :=
  ⋂ m, ⋃ i, ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))
μ : ProbabilityMeasure 𝓧
hs : μ ∈ S
m : ℕ
hbound : 1 - ε * 2 ^ (-↑(m + 1)) ≤ ↑(μ (⋃ i, ⋃ (_ : i ≤ km (m + 1)), ball (D i) (u (m + 1))))
⊢ 1 - ε * 2 ^ (-(↑m + 1)) ≤
    ↑μ (⋃ i, ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m)))

Tactic:
simp_all only [neg_add_rev, Int.reduceNeg, tsub_le_iff_right, Nat.cast_add, Nat.cast_one,
          ← coe_ofNat, ← ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure]

Goal state after:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝³ : PseudoMetricSpace 𝓧
inst✝² : OpensMeasurableSpace 𝓧
inst✝¹ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst✝ : 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), ⋃ i, ball (D i) (u m) = univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
km : ℕ → ℕ
bigK : Set 𝓧 :=
  ⋂ m, ⋃ i, ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))
μ : ProbabilityMeasure 𝓧
hs : μ ∈ S
m : ℕ
hbound : 1 ≤ ↑(μ (⋃ i, ⋃ (_ : i ≤ km (m + 1)), ball (D i) (u (m + 1)))) + ε * ↑2 ^ (-1 + -↑m)
⊢ 1 ≤
    ↑(μ
          (⋃ i,
            ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m)))) +
      ε * ↑2 ^ (-1 + -↑m)
Module: Mathlib.MeasureTheory.VectorMeasure.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
a : α
this✝ : Nonempty α
hlim : Tendsto f atTop (𝓝 (atTop.limUnder f))
u : ℕ → α
u_mono : Monotone u
hu : Tendsto u atTop atTop
A : Tendsto (fun n ↦ hf.vectorMeasure (Icc a (u n))) atTop (𝓝 (hf.vectorMeasure (Ici a)))
this : ∀ᶠ (n : ℕ) in atTop, a ≤ u n
n : ℕ
hn : a ≤ u n
⊢ Function.rightLim f (u n) - Function.leftLim f a = hf.vectorMeasure (Icc a (u n))

Tactic:
hf.vectorMeasure_Icc hn

Goal state after:
α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
a : α
this✝ : Nonempty α
hlim : Tendsto f atTop (𝓝 (atTop.limUnder f))
u : ℕ → α
u_mono : Monotone u
hu : Tendsto u atTop atTop
A : Tendsto (fun n ↦ hf.vectorMeasure (Icc a (u n))) atTop (𝓝 (hf.vectorMeasure (Ici a)))
this : ∀ᶠ (n : ℕ) in atTop, a ≤ u n
n : ℕ
hn : a ≤ u n
⊢ Function.rightLim f (u n) - Function.leftLim f a = Function.rightLim f (u n) - Function.leftLim f a
Module: Mathlib.MeasureTheory.VectorMeasure.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
a : α
this✝ : Nonempty α
hlim : Tendsto f atBot (𝓝 (atBot.limUnder f))
u : ℕ → α
u_anti : Antitone u
hu : Tendsto u atTop atBot
A : Tendsto (fun n ↦ hf.vectorMeasure (Icc (u n) a)) atTop (𝓝 (hf.vectorMeasure (Iic a)))
this : ∀ᶠ (n : ℕ) in atTop, u n ≤ a
n : ℕ
hn : u n ≤ a
⊢ Function.rightLim f a - Function.leftLim f (u n) = hf.vectorMeasure (Icc (u n) a)

Tactic:
hf.vectorMeasure_Icc hn

Goal state after:
α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
a : α
this✝ : Nonempty α
hlim : Tendsto f atBot (𝓝 (atBot.limUnder f))
u : ℕ → α
u_anti : Antitone u
hu : Tendsto u atTop atBot
A : Tendsto (fun n ↦ hf.vectorMeasure (Icc (u n) a)) atTop (𝓝 (hf.vectorMeasure (Iic a)))
this : ∀ᶠ (n : ℕ) in atTop, u n ≤ a
n : ℕ
hn : u n ≤ a
⊢ Function.rightLim f a - Function.leftLim f (u n) = Function.rightLim f a - Function.leftLim f (u n)