mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c : E
t : ℝ
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
ht : t < 1 / 2
instE : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
H₁ : ⇑(γab.trans γbc).extend =ᶠ[𝓝 t] fun s ↦ γab.extend (2 * s)
H₂ : 2 • I =ᶠ[𝓝 (2 * t)] I
⊢ (ω ((fun s ↦ γab.extend (2 * s)) t)) (derivWithin ⇑(γab.trans γbc).extend I t) = 2 • curveIntegralFun ω γab (2 * t)

Tactic:
H₁.derivWithin_eq_of_nhds,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c : E
t : ℝ
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
ht : t < 1 / 2
instE : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
H₁ : ⇑(γab.trans γbc).extend =ᶠ[𝓝 t] fun s ↦ γab.extend (2 * s)
H₂ : 2 • I =ᶠ[𝓝 (2 * t)] I
⊢ (ω ((fun s ↦ γab.extend (2 * s)) t)) (derivWithin (fun s ↦ γab.extend (2 * s)) I t) =
    2 • curveIntegralFun ω γab (2 * t)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case left
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 : s₀ᶜ ∪ s₁ᶜ = univ
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)
x : X
hx : x ∈ t
h : x ∈ s₀ᶜ ∨ x ∈ s₁ᶜ
⊢ x ∈ s₀ᶜ → ∃ i, x ∈ if i = 0 then s₀ᶜ else s₁ᶜ

pp: case right
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 : s₀ᶜ ∪ s₁ᶜ = univ
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)
x : X
hx : x ∈ t
h : x ∈ s₀ᶜ ∨ x ∈ s₁ᶜ
⊢ x ∈ s₁ᶜ → ∃ i, x ∈ if i = 0 then s₀ᶜ else s₁ᶜ

Tactic:
· intro h0
      use 0
      simp only [Fin.isValue, ↓reduceIte]
      exact h0

Goal state after:
case right
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 : s₀ᶜ ∪ s₁ᶜ = univ
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)
x : X
hx : x ∈ t
h : x ∈ s₀ᶜ ∨ x ∈ s₁ᶜ
⊢ x ∈ s₁ᶜ → ∃ i, x ∈ if i = 0 then s₀ᶜ else s₁ᶜ
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
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑(g₁ + g₂)) 1 (tsupport f.toFun)
x : X
⊢ ↑(f x) = ↑((⇑g₁ + ⇑g₂) x) * ↑(f x)

Tactic:
by_cases h : f x = 0

Goal state after:
case pos
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
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑(g₁ + g₂)) 1 (tsupport f.toFun)
x : X
h : f x = 0
⊢ ↑(f x) = ↑((⇑g₁ + ⇑g₂) x) * ↑(f x)

case neg
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
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑(g₁ + g₂)) 1 (tsupport f.toFun)
x : X
h : ¬f x = 0
⊢ ↑(f x) = ↑((⇑g₁ + ⇑g₂) x) * ↑(f x)
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)
⊢ b ≤ ↑(rieszContentAux Λ K)

Tactic:
rieszContentAux,

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)
⊢ b ≤ ↑(sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x}))
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 < rieszContentAux Λ K + (↑ε).toNNReal
⊢ ↑b.toNNReal ≤ ↑(sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x})) + ↑ε

Tactic:
rieszContentAux,

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}) + (↑ε).toNNReal
⊢ ↑b.toNNReal ≤ ↑(sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x})) + ↑ε
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
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
⊢ (rieszMeasure Λ) K ≤ ENNReal.ofReal (Λ f)

Tactic:
rw [← Compacts.coe_mk K hK, rieszMeasure,
    Content.measure_eq_content_of_regular _ (contentRegular_rieszContent (toNNRealLinear Λ))]
  apply ENNReal.coe_le_iff.mpr
  intro p hp
  rw [← ENNReal.ofReal_coe_nnreal,
    ENNReal.ofReal_eq_ofReal_iff (Λ.map_nonneg hf) NNReal.zero_le_coe] at hp
  apply csInf_le'
  rw [Set.mem_image]
  use f.nnrealPart
  simp_rw [Set.mem_ofPred_eq, nnrealPart_apply, Real.one_le_toNNReal]
  refine ⟨(fun x hx ↦ Eq.ge (hfK x hx)), ?_⟩
  apply NNReal.eq
  rw [toNNRealLinear_apply, show f.nnrealPart.toReal = f by ext z; simp [hf z], hp]

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: case iUnion
α : Type u
s : Set (Set α)
t : Set α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ generateMeasurableRec s (ω_ 1)
⊢ ⋃ i, f i ∈ generateMeasurableRec s (ω_ 1)

Tactic:
simp_rw [generateMeasurableRec_omega_one, mem_iUnion₂, exists_prop] at IH
    exact iUnion_mem_generateMeasurableRec IH

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

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
t : ℝ
⊢ taylorWithinEval (charFun μ) n univ 0 t = ∑ k ∈ Finset.range (n + 1), (↑k !)⁻¹ * (↑t * I) ^ k * ↑(∫ (x : ℝ), x ^ k ∂μ)

Tactic:
by
  simp_rw [taylor_within_apply, sub_zero, RCLike.real_smul_eq_coe_mul]
  refine Finset.sum_congr rfl fun k hkn ↦ ?_
  push_cast
  have hint' : MemLp id k μ := hint.mono_exponent (by simp_all)
  simp [iteratedDeriv_charFun_zero hint', mul_pow, mul_comm, mul_assoc, mul_left_comm]

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

Goal state before:
pp: ι : Type u_6
inst✝⁷ : Fintype ι
E : ι → Type u_7
inst✝⁶ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁵ : (i : ι) → InnerProductSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
inst✝⁴ : ∀ (i : ι), CompleteSpace (E i)
inst✝³ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
μ : (i : ι) → Measure (E i)
ν : Measure ((i : ι) → E i)
inst✝¹ : ∀ (i : ι), IsFiniteMeasure (μ i)
inst✝ : IsFiniteMeasure ν
h : ν = Measure.pi μ
⊢ ∀ (t : WithLp 2 ((i : ι) → E i)), charFun (Measure.map (toLp 2) (Measure.pi μ)) t = ∏ i, charFun (μ i) (t.ofLp i)

Tactic:
exact charFun_pi

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

Goal state before:
pp: E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
μ ν : Measure E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
L : StrongDual ℝ E
⊢ charFunDual (μ ∗ ν) L = charFunDual μ L * charFunDual ν L

Tactic:
simp_rw [charFunDual_apply]

Goal state after:
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
μ ν : Measure E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
L : StrongDual ℝ E
⊢ ∫ (v : E), cexp (↑(L v) * I) ∂μ ∗ ν = (∫ (v : E), cexp (↑(L v) * I) ∂μ) * ∫ (v : E), cexp (↑(L v) * I) ∂ν
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:
Integrable.integral_eq_integral_Ioc_meas_le (M := ‖f‖) ?_ f_nn ?_

Goal state after:
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
⊢ ∫ (t : ℝ) in Ioc 0 ‖f‖, μ.real {a | t ≤ f a} = ∫ (t : ℝ) in Ioc 0 ‖f‖, μ.real {a | t ≤ f a}

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
⊢ Integrable (⇑f) μ

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
⊢ ⇑f ≤ᵐ[μ] fun x ↦ ‖f‖
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
t : ℝ
⊢ (fun t ↦ μ.real {a | t ≤ f a}) t ≤ (fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε) t

Tactic:
simp only [measureReal_def]

Goal state after:
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
t : ℝ
⊢ (μ {a | t ≤ f a}).toReal ≤ (ν (thickening ε {a | t ≤ f a})).toReal + ε
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case f_mble
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
⊢ AEStronglyMeasurable (fun t ↦ μ.real {a | t ≤ f a}) volume

pp: case f_bdd
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ioc 0 ‖f‖), ‖μ.real {a_1 | a ≤ f a_1}‖ ≤ μ.real univ

Tactic:
· apply (Measurable.ennreal_toReal (Antitone.measurable ?_)).aestronglyMeasurable
      exact fun _ _ hst ↦ measure_mono (fun _ h ↦ hst.trans h)

Goal state after:
case f_bdd
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ioc 0 ‖f‖), ‖μ.real {a_1 | a ≤ f a_1}‖ ≤ μ.real univ
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureProd

Goal state before:
pp: α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
μ : ProbabilityMeasure α × ProbabilityMeasure β
S : Set (Set (α × β)) :=
  {t | ∃ a b, MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0 ∧ t = a ×ˢ b}
this✝¹ : IsPiSystem S
this✝ : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
u : Set (α × β)
u_open : IsOpen[instTopologicalSpaceProd] u
x : α × β
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ℝ
hr : r ∈ Ioo 0 ε
r' : ℝ
hr' : r' ∈ Ioo 0 ε
μr : ↑μ.1 (frontier (ball x.1 r)) = 0
μr' : ↑μ.2 (frontier (ball x.2 r')) = 0
⊢ μ.1 (frontier (ball x.1 r)) = 0

Tactic:
by simp [coeFn_def, μr]

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
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun i ↦ ∫ (x : Ω), f x ∂↑(Ps i)

Tactic:
simp only [IsCoboundedUnder, IsCobounded, eventually_map, eventually_atTop,
               forall_exists_index]
    refine ⟨0, fun a i hia ↦ le_trans (integral_nonneg f_nn) (hia i le_rfl)⟩

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))
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist (ofMeasure x) (ofMeasure P) < ε

Tactic:
have mem_nhds_P (G : Set Ω) (G_open : IsOpen G) :
      {Q | P.toMeasure G < Q.toMeasure G + ENNReal.ofReal (ε/3)} ∈ 𝓝 P :=
    P.toMeasure_add_pos_gt_mem_nhds G_open third_ε_pos'

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
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist (ofMeasure x) (ofMeasure P) < ε
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹⁸ : Group A
inst✝¹⁷ : Group B
inst✝¹⁶ : Group C
inst✝¹⁵ : TopologicalSpace A
inst✝¹⁴ : TopologicalSpace B
inst✝¹³ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹² : IsTopologicalGroup A
inst✝¹¹ : IsTopologicalGroup B
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : IsTopologicalGroup C
inst✝⁵ : LocallyCompactSpace B
inst✝⁴ : MeasurableSpace C
inst✝³ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝² : T2Space B
inst✝¹ : MeasurableSpace B
inst✝ : BorelSpace B
U : Set B
hU : IsOpen U
b : B
hb✝ : b ∈ U
K : Set B
hK : IsCompact K
hb : b ∈ interior K
hKU : K ⊆ U
f : C(B, ℝ)
hf1 : Set.EqOn (⇑f) 1 K
hf2 : IsCompact (tsupport ⇑f)
hf3 : tsupport ⇑f ⊆ U
hf4 : ∀ (x : B), f x ∈ Set.Icc 0 1
hf0 : 0 ≤ (H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }
⊢ (H.inducedMeasure μA μC) U ≠ 0

Tactic:
← pos_iff_ne_zero,

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹⁸ : Group A
inst✝¹⁷ : Group B
inst✝¹⁶ : Group C
inst✝¹⁵ : TopologicalSpace A
inst✝¹⁴ : TopologicalSpace B
inst✝¹³ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹² : IsTopologicalGroup A
inst✝¹¹ : IsTopologicalGroup B
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : IsTopologicalGroup C
inst✝⁵ : LocallyCompactSpace B
inst✝⁴ : MeasurableSpace C
inst✝³ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝² : T2Space B
inst✝¹ : MeasurableSpace B
inst✝ : BorelSpace B
U : Set B
hU : IsOpen U
b : B
hb✝ : b ∈ U
K : Set B
hK : IsCompact K
hb : b ∈ interior K
hKU : K ⊆ U
f : C(B, ℝ)
hf1 : Set.EqOn (⇑f) 1 K
hf2 : IsCompact (tsupport ⇑f)
hf3 : tsupport ⇑f ⊆ U
hf4 : ∀ (x : B), f x ∈ Set.Icc 0 1
hf0 : 0 ≤ (H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }
⊢ 0 < (H.inducedMeasure μA μC) U
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case e'_3
ι : Type u_1
inst✝¹ : Nonempty ι
inst✝ : Fintype ι
x : EuclideanSpace ℝ ι
r : ℝ
hr : 0 ≤ r
⊢ √π = 2 * Gamma (1 / 2 + 1)

Tactic:
Gamma_add_one (by simp),

Goal state after:
case e'_3
ι : Type u_1
inst✝¹ : Nonempty ι
inst✝ : Fintype ι
x : EuclideanSpace ℝ ι
r : ℝ
hr : 0 ≤ r
⊢ √π = 2 * (1 / 2 * Gamma (1 / 2))
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝ : NormedAddCommGroup E
hS : IsTightMeasureSet S
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)

Tactic:
have h := tendsto_measure_compl_closedBall_of_isTightMeasureSet hS 0
  convert! h using 6 with r
  ext
  simp

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

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : ProperSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : IsTightMeasureSet (Set.range μ)
h_sup : Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
⊢ Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖x‖}) atTop) atTop (𝓝 0)

Tactic:
refine tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds h_sup (fun _ ↦ zero_le) ?_

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : ProperSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : IsTightMeasureSet (Set.range μ)
h_sup : Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
⊢ (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖x‖}) atTop) ≤ fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖x‖}
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_2
ι : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype ι
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis ι 𝕜 E
h : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪b i, x⟫_𝕜‖}) atTop (𝓝 0)
hE : Nontrivial E
h_rank : 0 < ↑(Fintype.card ι)
this✝ : Nonempty ι
this : ProperSpace E
r : ℝ
⊢ ⨆ μ ∈ S, μ (⋃ i, {x | r / √↑(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖}) ≤
    ⨆ μ ∈ S, ∑ i, μ {x | r / √↑(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖}

Tactic:
gcongr with μ hμS
      exact measure_iUnion_fintype_le μ _

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

Goal state before:
pp: case hbc.refine_2
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
⊢ (fun x ↦ ‖1 - charFun (μ n) (x • z)‖) ≤ᵐ[volume.restrict (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹))] fun x ↦ 2

Tactic:
· exact ae_of_all _ fun _ ↦ norm_one_sub_charFun_le_two

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h : mα = generateFrom C
s✝ : Set α
hs✝ : MeasurableSet s✝
ε✝ : ℝ≥0∞
hε : 0 < ε✝
s : Set α
hs : MeasurableSet s
h's : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ C, μ (t ∆ s) < ε
ε : ℝ≥0∞
εpos : 0 < ε
t : Set α
tC : t ∈ C
ht : μ (t ∆ s) < ε / 2
D : Set (Set α)
D_count : D.Countable
DC : D ⊆ C
hD : μ (⋃₀ D)ᶜ = 0
⊢ μ (⋃₀ (D ∪ {∅}))ᶜ = 0 ∧ (D ∪ {∅}).Nonempty

Tactic:
simp only [union_singleton, sUnion_insert, empty_union, insert_nonempty, and_true, hD]

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

Goal state before:
pp: case refine_1
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))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
⊢ μ.mass ≤ C

Tactic:
simp only [mass, mk_apply, μ]
    rw [show C = (C : ℝ≥0∞).toNNReal by simp]
    exact ENNReal.toNNReal_mono (by simp) B

Goal state after: