mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: a' b' ε : ℝ
hε : 0 < ε
⊢ ∃ N, 0 < N ∧ a' / ↑N * (b' + a' / ↑N) ≤ ε

Tactic:
have A : Tendsto (fun (N : ℝ) ↦ a' / N * (b' + a' / N)) atTop (𝓝 (0 * (b' + 0))) := by
    apply Tendsto.mul
    · exact Tendsto.div_atTop tendsto_const_nhds tendsto_id
    · exact Tendsto.add tendsto_const_nhds (Tendsto.div_atTop tendsto_const_nhds tendsto_id)
  have B := A.comp tendsto_natCast_atTop_atTop
  simp only [add_zero, zero_mul] at B
  obtain ⟨N, hN, h'N⟩ := (((tendsto_order.1 B).2 _ hε).and (Ici_mem_atTop 1)).exists
  exact ⟨N, h'N, hN.le⟩

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: case refine_1
E : Type u_3
inst✝⁷ : MeasurableSpace E
μ ν : Measure E
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : CompleteSpace E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ∀ (x : E), ∫ (v : E), (innerProbChar x) v ∂μ = ∫ (v : E), (innerProbChar x) v ∂ν
⊢ ∀ (v : E), v ≠ 0 → (innerₗ E) v ≠ 0

pp: case refine_2
E : Type u_3
inst✝⁷ : MeasurableSpace E
μ ν : Measure E
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : CompleteSpace E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ∀ (x : E), ∫ (v : E), (innerProbChar x) v ∂μ = ∫ (v : E), (innerProbChar x) v ∂ν
⊢ Continuous fun p ↦ ((innerₗ E) p.1) p.2

Tactic:
· exact fun v hv ↦ DFunLike.ne_iff.mpr ⟨v, inner_self_ne_zero.mpr hv⟩

Goal state after:
case refine_2
E : Type u_3
inst✝⁷ : MeasurableSpace E
μ ν : Measure E
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : CompleteSpace E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ∀ (x : E), ∫ (v : E), (innerProbChar x) v ∂μ = ∫ (v : E), (innerProbChar x) v ∂ν
⊢ Continuous fun p ↦ ((innerₗ E) p.1) p.2
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 : ↥I' → Set X
hV : ∀ (i : ↥I'), ∃ t_1 ⊆ V i, IsOpen[t ↑i] t_1 ∧ x ∈ t_1
⊢ ∃ f, Continuous[⨅ i, t i, _] f ∧ f x = 0 ∧ EqOn f 1 (⋂ i, V i)ᶜ

Tactic:
choose U hUV hU hxU using hV

Goal state after:
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
hU : ∀ (i : ↥I'), IsOpen[t ↑i] (U i)
hxU : ∀ (i : ↥I'), x ∈ U i
⊢ ∃ f, Continuous[⨅ i, t i, _] f ∧ f x = 0 ∧ EqOn f 1 (⋂ i, V i)ᶜ
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: case a
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 : Uᶜ ⊆ f ⁻¹' {1}
⊢ ∃ i, stoneCechUnit x ∈ i ∧ IsOpen i ∧ stoneCechUnit ⁻¹' i ⊆ U

Tactic:
← compl_subset_comm,

Goal state after:
case a
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 : (f ⁻¹' {1})ᶜ ⊆ U
⊢ ∃ i, stoneCechUnit x ∈ i ∧ IsOpen i ∧ stoneCechUnit ⁻¹' i ⊆ U
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
f : Ω →ᵇ ℝ
A : Set ℝ
A_finmeas : volume A ≠ ∞
μ : ProbabilityMeasure Ω
n : ℝ
⊢ Measurable fun t ↦ ↑(μ (thickening n {a | t ≤ f a}))

Tactic:
simp only [measurable_coe_nnreal_real_iff]

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
f : Ω →ᵇ ℝ
A : Set ℝ
A_finmeas : volume A ≠ ∞
μ : ProbabilityMeasure Ω
n : ℝ
⊢ Measurable fun x ↦ μ (thickening n {a | x ≤ f a})
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:

Tactic:
μ.real K = ∫ x, K.indicator 1 x ∂μ := integral_indicator_one hK.measurableSet |>.symm
    _ ≤ ∫ x, f x ∂μ := by
      refine integral_mono ?_ f.integrable hfK
      exact (continuousOn_const.integrableOn_compact hK).integrable_indicator hK.measurableSet
    _ ≤ ∫ x, f x ∂ν := hμν f
    _ ≤ ∫ x, V.indicator 1 x ∂ν := by
      refine integral_mono f.integrable ?_ hfV
      exact IntegrableOn.integrable_indicator integrableOn_const pV2.measurableSet
    _ ≤ (ν K).toReal + ↑ε := by
      rwa [integral_indicator_one pV2.measurableSet, measureReal_def,
        ← ENNReal.coe_toReal, ← ENNReal.toReal_add, ENNReal.toReal_le_toReal]
      all_goals finiteness

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)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω),
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
hQ :
  ∀ i ⊆ Iio N,
    ↑P (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) < ↑Q (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) + ENNReal.ofReal (ε / 3)
⊢ dist (ofMeasure Q) (ofMeasure P) ≤ 2 * (ε / 3)

Tactic:
rw [dist_comm, dist_probabilityMeasure_def]

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)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω),
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
hQ :
  ∀ i ⊆ Iio N,
    ↑P (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) < ↑Q (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) + ENNReal.ofReal (ε / 3)
⊢ levyProkhorovDist ↑(ofMeasure P).toMeasure ↑(ofMeasure Q).toMeasure ≤ 2 * (ε / 3)
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
⊢ ∫ (t : ℝ) in -r..r, charFun μ t = 2 * ↑r * ↑(∫ (x : ℝ), sinc (r * x) ∂μ)

Tactic:
calc ∫ t in -r..r, charFun μ t
  _ = ∫ x in -r..r, ∫ y, cexp (x * y * I) ∂μ := by simp_rw [charFun_apply_real]
  _ = ∫ y, ∫ x in -r..r, cexp (x * y * I) ∂volume ∂μ := by
    rw [intervalIntegral_integral_swap h_int]
  _ = ∫ y, if r * y = 0 then 2 * (r : ℂ)
      else y⁻¹ * ∫ x in -(y * r)..y * r, cexp (x * I) ∂volume ∂μ := by
    congr with y
    by_cases hy : y = 0
    · simp [hy, two_mul]
    simp only [mul_eq_zero, hr.ne', hy, or_self, ↓reduceIte, ofReal_inv]
    have h := intervalIntegral.integral_deriv_smul_comp (E := ℂ) (a := -r) (b := r)
      (f := fun x ↦ y * x) (f' := fun _ ↦ y) (g := fun x ↦ cexp (x * I)) ?_ (by fun_prop)
      (by fun_prop)
    swap
    · intro x hx
      simp_rw [mul_comm y]
      exact hasDerivAt_mul_const _
    simp only [Function.comp_apply, ofReal_mul, real_smul, intervalIntegral.integral_const_mul,
      mul_neg] at h
    rw [← h, ← mul_assoc]
    norm_cast
    simp [mul_comm _ y, mul_inv_cancel₀ hy]
  _ = ∫ x, 2 * (r : ℂ) * sinc (r * x) ∂μ := by
    congr with y
    rw [integral_exp_mul_I_eq_sinc]
    split_ifs with hry
    · simp [hry]
    have hy : y ≠ 0 := fun hy ↦ hry (by simp [hy])
    norm_cast
    field_simp
  _ = 2 * r * ∫ x, sinc (r * x) ∂μ := by
    norm_cast
    rw [← integral_const_mul]

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

Goal state before:
pp: 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
⊢ (fun μ ↦ ∫ (x : E), g x ∂↑μ) ⁻¹'
      Icc (-(↑C * ‖g.toBoundedContinuousFunction‖)) (↑C * ‖g.toBoundedContinuousFunction‖) ∈
    f

Tactic:
filter_upwards [hf] with μ hμ

Goal state after:
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
⊢ μ ∈
    (fun μ ↦ ∫ (x : E), g x ∂↑μ) ⁻¹'
      Icc (-(↑C * ‖g.toBoundedContinuousFunction‖)) (↑C * ‖g.toBoundedContinuousFunction‖)
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: 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))
g g' : E →C_c ℝ
⊢ Λ (g + g') = Λ g + Λ g'

Tactic:
by
      have : Tendsto (fun (μ : FiniteMeasure E) ↦ ∫ x, g x + g' x ∂μ) f (𝓝 (Λ g + Λ g')) := by
        convert! (hΛ g).add (hΛ g')
        rw [integral_add]
        · exact g.continuous.integrable_of_hasCompactSupport g.hasCompactSupport
        · exact g'.continuous.integrable_of_hasCompactSupport g'.hasCompactSupport
      exact tendsto_nhds_unique (hΛ (g + g')) this

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case e'_3.e'_5
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p
eq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℝ), (∑ i, |x i| ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ i, |0 i| ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℝ), (∑ i, |(-x) i| ^ p) ^ (1 / p) = (∑ i, |x i| ^ p) ^ (1 / p)
nm_add : ∀ (x y : ι → ℝ), (∑ i, |(x + y) i| ^ p) ^ (1 / p) ≤ (∑ i, |x i| ^ p) ^ (1 / p) + (∑ i, |y i| ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℝ), (∑ i, |(r • x) i| ^ p) ^ (1 / p) ≤ |r| * (∑ i, |x i| ^ p) ^ (1 / p)
⊢ (2 * Gamma (1 / p + 1)) ^ card ι = ∫ (x : ι → ℝ), rexp (-((∑ i, |x i| ^ p) ^ (1 / p)) ^ p)

pp: case e'_3.e'_6
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p
eq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℝ), (∑ i, |x i| ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ i, |0 i| ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℝ), (∑ i, |(-x) i| ^ p) ^ (1 / p) = (∑ i, |x i| ^ p) ^ (1 / p)
nm_add : ∀ (x y : ι → ℝ), (∑ i, |(x + y) i| ^ p) ^ (1 / p) ≤ (∑ i, |x i| ^ p) ^ (1 / p) + (∑ i, |y i| ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℝ), (∑ i, |(r • x) i| ^ p) ^ (1 / p) ≤ |r| * (∑ i, |x i| ^ p) ^ (1 / p)
⊢ ↑(card ι) / p + 1 = ↑(finrank ℝ (ι → ℝ)) / p + 1

Tactic:
· simp_rw [← rpow_mul (h₂ _), div_mul_cancel₀ _ (ne_of_gt h₁), Real.rpow_one,
      ← Finset.sum_neg_distrib, exp_sum]
    rw [integral_fintype_prod_volume_eq_pow fun x : ℝ => exp (- |x| ^ p), integral_comp_abs
      (f := fun x => exp (- x ^ p)), integral_exp_neg_rpow h₁]

Goal state after:
case e'_3.e'_6
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p
eq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℝ), (∑ i, |x i| ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ i, |0 i| ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℝ), (∑ i, |(-x) i| ^ p) ^ (1 / p) = (∑ i, |x i| ^ p) ^ (1 / p)
nm_add : ∀ (x y : ι → ℝ), (∑ i, |(x + y) i| ^ p) ^ (1 / p) ≤ (∑ i, |x i| ^ p) ^ (1 / p) + (∑ i, |y i| ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℝ), (∑ i, |(r • x) i| ^ p) ^ (1 / p) ≤ |r| * (∑ i, |x i| ^ p) ^ (1 / p)
⊢ ↑(card ι) / p + 1 = ↑(finrank ℝ (ι → ℝ)) / p + 1
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
this✝¹ : Tendsto (fun n ↦ ∑ i ∈ Finset.range (n + 1), ↑(ν i) univ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) univ))
n : ℕ
this✝ : ∑ i ∈ Finset.range (n + 1), ↑(ν i) univ = ↑(∑ i ∈ Finset.range (n + 1), ν i) univ
this :
  Tendsto (fun μ ↦ (∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i)).mass) (↑f)
    (𝓝 (∑ i ∈ Finset.range (n + 1), ν i).mass)
⊢ ∀ᶠ (c : FiniteMeasure E) in ↑f, (∑ i ∈ Finset.range (n + 1), c.restrict (disjointed K i)).mass ≤ C

Tactic:
filter_upwards [hf] with μ hμ

Goal state after:
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
this✝¹ : Tendsto (fun n ↦ ∑ i ∈ Finset.range (n + 1), ↑(ν i) univ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) univ))
n : ℕ
this✝ : ∑ i ∈ Finset.range (n + 1), ↑(ν i) univ = ↑(∑ i ∈ Finset.range (n + 1), ν i) univ
this :
  Tendsto (fun μ ↦ (∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i)).mass) (↑f)
    (𝓝 (∑ i ∈ Finset.range (n + 1), ν i).mass)
μ : FiniteMeasure E
hμ : μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n
⊢ (∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i)).mass ≤ C
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case refine_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
hφs : range ⇑φ ⊆ s
⊢ ∀ a_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, φ (a_1, b_1) ∈ range ⇑φ

Tactic:
intros
    apply mem_range_self

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

Goal state before:
pp: 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
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤
    2⁻¹ * r * ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, ‖1 - charFun (μ n) (t • z)‖

Tactic:
by
      grw [neg_mul, intervalIntegral.norm_integral_le_integral_norm hr']

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

Goal state before:
pp: case hgi
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⋯
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : NormalSpace E
this✝ : (↑μ).InnerRegular
g : E → ℝ
g_cont : Continuous[inst✝², _] g
gK : EqOn g 0 (K n)
g_nonneg : 0 ≤ g
g_le : g ≤ 1
this : Tendsto (fun ρ ↦ ∫ (x : E), g x ∂↑ρ) (↑f) (𝓝 (∫ (x : E), g x ∂↑μ))
ρ : FiniteMeasure E
hρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), ρ (K n)ᶜ ≤ u n
⊢ Integrable ((K n)ᶜ.indicator 1) ↑ρ

Tactic:
apply Integrable.indicator (integrable_const _) (hK n).measurableSet.compl

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

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
⊢ preVariationFun f ∅ = 0

Tactic:
simp [preVariationFun]

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

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
x✝ : Set X
⊢ preVariationFun 0 x✝ = 0 x✝

Tactic:
simp [preVariationFun]

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

Goal state before:
pp: κ : Type u_2
Z : Type u_3
Y : κ → Type u_6
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁸ : (j : κ) → TopologicalSpace (Y j)
inst✝⁷ : ∀ (j : κ), BorelSpace (Y j)
inst✝⁶ : ∀ (j : κ), HasOuterApproxClosed (Y j)
mZ : MeasurableSpace Z
inst✝⁵ : TopologicalSpace Z
inst✝⁴ : BorelSpace Z
inst✝³ : HasOuterApproxClosed Z
inst✝² : Finite κ
μ ν : Measure (Z × ((i : κ) → Y i))
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : Z →ᵇ ℝ) (g : ((j : κ) → Y j) →ᵇ ℝ),
    ∫ (p : Z × ((i : κ) → Y i)), f p.1 * g p.2 ∂μ = ∫ (p : Z × ((i : κ) → Y i)), f p.1 * g p.2 ∂ν
this : Fintype κ
f : Z →ᵇ ℝ
g : (j : κ) → Y j →ᵇ ℝ
⊢ ∫ (p : Z × ((j : κ) → Y j)), f p.1 * ∏ j, (g j) (p.2 j) ∂μ =
    ∫ (p : Z × ((j : κ) → Y j)), f p.1 * ∏ j, (g j) (p.2 j) ∂ν

Tactic:
convert! h f (∏ j, (g j).compContinuous ⟨Function.eval j, by fun_prop⟩)

Goal state after:
case e'_2
κ : Type u_2
Z : Type u_3
Y : κ → Type u_6
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁸ : (j : κ) → TopologicalSpace (Y j)
inst✝⁷ : ∀ (j : κ), BorelSpace (Y j)
inst✝⁶ : ∀ (j : κ), HasOuterApproxClosed (Y j)
mZ : MeasurableSpace Z
inst✝⁵ : TopologicalSpace Z
inst✝⁴ : BorelSpace Z
inst✝³ : HasOuterApproxClosed Z
inst✝² : Finite κ
μ ν : Measure (Z × ((i : κ) → Y i))
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : Z →ᵇ ℝ) (g : ((j : κ) → Y j) →ᵇ ℝ),
    ∫ (p : Z × ((i : κ) → Y i)), f p.1 * g p.2 ∂μ = ∫ (p : Z × ((i : κ) → Y i)), f p.1 * g p.2 ∂ν
this : Fintype κ
f : Z →ᵇ ℝ
g : (j : κ) → Y j →ᵇ ℝ
x✝ : Z × ((j : κ) → Y j)
⊢ ∏ j, (g j) (x✝.2 j) = (∏ j, (g j).compContinuous { toFun := Function.eval j, continuous_toFun := ⋯ }) x✝.2

case e'_3
κ : Type u_2
Z : Type u_3
Y : κ → Type u_6
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁸ : (j : κ) → TopologicalSpace (Y j)
inst✝⁷ : ∀ (j : κ), BorelSpace (Y j)
inst✝⁶ : ∀ (j : κ), HasOuterApproxClosed (Y j)
mZ : MeasurableSpace Z
inst✝⁵ : TopologicalSpace Z
inst✝⁴ : BorelSpace Z
inst✝³ : HasOuterApproxClosed Z
inst✝² : Finite κ
μ ν : Measure (Z × ((i : κ) → Y i))
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : Z →ᵇ ℝ) (g : ((j : κ) → Y j) →ᵇ ℝ),
    ∫ (p : Z × ((i : κ) → Y i)), f p.1 * g p.2 ∂μ = ∫ (p : Z × ((i : κ) → Y i)), f p.1 * g p.2 ∂ν
this : Fintype κ
f : Z →ᵇ ℝ
g : (j : κ) → Y j →ᵇ ℝ
x✝ : Z × ((j : κ) → Y j)
⊢ ∏ j, (g j) (x✝.2 j) = (∏ j, (g j).compContinuous { toFun := Function.eval j, continuous_toFun := ⋯ }) x✝.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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⋯
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : Monotone K
this✝¹ : Tendsto (fun m ↦ ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) (K n)ᶜ))
m : ℕ
hm : n ≤ m
this✝ : ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ = ∑ i ∈ Finset.Ioc n m, ↑(ν i) (K n)ᶜ
this :
  Tendsto (fun μ ↦ (∑ i ∈ Finset.Ioc n m, μ.restrict (disjointed K i)).mass) (↑f) (𝓝 (∑ i ∈ Finset.Ioc n m, ν i).mass)
⊢ ∀ᶠ (c : FiniteMeasure E) in ↑f, (∑ i ∈ Finset.Ioc n m, c.restrict (disjointed K i)).mass ≤ u n

Tactic:
filter_upwards [hf] with μ hμ

Goal state after:
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ✝ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ✝)
n : ℕ
h : Monotone K
this✝¹ : Tendsto (fun m ↦ ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) (K n)ᶜ))
m : ℕ
hm : n ≤ m
this✝ : ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ = ∑ i ∈ Finset.Ioc n m, ↑(ν i) (K n)ᶜ
this :
  Tendsto (fun μ ↦ (∑ i ∈ Finset.Ioc n m, μ.restrict (disjointed K i)).mass) (↑f) (𝓝 (∑ i ∈ Finset.Ioc n m, ν i).mass)
μ : FiniteMeasure E
hμ : μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n
⊢ (∑ i ∈ Finset.Ioc n m, μ.restrict (disjointed K i)).mass ≤ u n
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: case inl.inl.inl
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
this✝ : μ (s ∩ t) ≤ μ s
this : μ (s ∩ t) ≤ μ t
h✝² : μ s = 0
h✝¹ : μ t = 0
h✝ : μ (s ∩ t) = 0
⊢ μ (s ∩ t) = μ s * μ t

Tactic:
simp_all [measure_inter_eq_one]

Goal state after: