mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: 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)
nm_zero : ‖toLp (ENNReal.ofReal p) 0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = 0 ↔ x = 0
nm_neg : ∀ (x : ι → ℝ), ‖-toLp (ENNReal.ofReal p) x‖ = ‖toLp (ENNReal.ofReal p) x‖
nm_add :
  ∀ (x y : ι → ℝ),
    ‖toLp (ENNReal.ofReal p) x + toLp (ENNReal.ofReal p) y‖ ≤ ‖toLp (ENNReal.ofReal p) x‖ + ‖toLp (ENNReal.ofReal p) y‖
⊢ volume {x | ∑ i, |x i| ^ p < 1} = ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))

Tactic:
have nm_smul := fun (r : ℝ) (x : ι → ℝ) =>
    norm_smul_le (β := PiLp (.ofReal p) (fun _ : ι => ℝ)) r (toLp (.ofReal p) x)

Goal state after:
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)
nm_zero : ‖toLp (ENNReal.ofReal p) 0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = 0 ↔ x = 0
nm_neg : ∀ (x : ι → ℝ), ‖-toLp (ENNReal.ofReal p) x‖ = ‖toLp (ENNReal.ofReal p) x‖
nm_add :
  ∀ (x y : ι → ℝ),
    ‖toLp (ENNReal.ofReal p) x + toLp (ENNReal.ofReal p) y‖ ≤ ‖toLp (ENNReal.ofReal p) x‖ + ‖toLp (ENNReal.ofReal p) y‖
nm_smul : ∀ (r : ℝ) (x : ι → ℝ), ‖r • toLp (ENNReal.ofReal p) x‖ ≤ ‖r‖ * ‖toLp (ENNReal.ofReal p) x‖
⊢ volume {x | ∑ i, |x i| ^ p < 1} = ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))
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))
c : ℝ
g : E →C_c ℝ
this : Tendsto (fun μ ↦ ∫ (x : E), c • g x ∂↑μ) (↑f) (𝓝 (c • Λ g))
⊢ Λ (c • g) = (RingHom.id ℝ) c • Λ g

Tactic:
exact tendsto_nhds_unique (hΛ (c • g)) this

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

Goal state before:
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:
finrank_fintype_fun_eq_card

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 = ↑(card ι) / p + 1
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
eq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖toLp (ENNReal.ofReal p) 0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = 0 ↔ x = 0
nm_neg : ∀ (x : ι → ℝ), ‖-toLp (ENNReal.ofReal p) x‖ = ‖toLp (ENNReal.ofReal p) x‖
nm_add :
  ∀ (x y : ι → ℝ),
    ‖toLp (ENNReal.ofReal p) x + toLp (ENNReal.ofReal p) y‖ ≤ ‖toLp (ENNReal.ofReal p) x‖ + ‖toLp (ENNReal.ofReal p) y‖
⊢ volume {x | (∑ i, |x i| ^ p) ^ (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))

Tactic:
have nm_smul := fun (r : ℝ) (x : ι → ℝ) =>
    norm_smul_le (β := PiLp (.ofReal p) (fun _ : ι => ℝ)) r (toLp (.ofReal p) x)

Goal state after:
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
eq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖toLp (ENNReal.ofReal p) 0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = 0 ↔ x = 0
nm_neg : ∀ (x : ι → ℝ), ‖-toLp (ENNReal.ofReal p) x‖ = ‖toLp (ENNReal.ofReal p) x‖
nm_add :
  ∀ (x y : ι → ℝ),
    ‖toLp (ENNReal.ofReal p) x + toLp (ENNReal.ofReal p) y‖ ≤ ‖toLp (ENNReal.ofReal p) x‖ + ‖toLp (ENNReal.ofReal p) y‖
nm_smul : ∀ (r : ℝ) (x : ι → ℝ), ‖r • toLp (ENNReal.ofReal p) x‖ ≤ ‖r‖ * ‖toLp (ENNReal.ofReal p) x‖
⊢ volume {x | (∑ i, |x i| ^ p) ^ (1 / p) ≤ r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd

Goal state before:
pp: case h1
ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁵ : (j : κ) → TopologicalSpace (Y j)
inst✝⁴ : ∀ (j : κ), BorelSpace (Y j)
inst✝³ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝² : Fintype ι
inst✝¹ : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝ : IsFiniteMeasure μ
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ≥0) (g : (j : κ) → Y j →ᵇ ℝ≥0),
    ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂μ =
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂ν
hμν : μ univ = ν univ
this✝ : IsFiniteMeasure ν
π : Set (Set (((i : ι) → X i) × ((j : κ) → Y j))) :=
  image2 (fun s t ↦ s ×ˢ t) (univ.pi '' univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s})
    (univ.pi '' univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t})
hπ1 : IsPiSystem π
hπ2 : Prod.instMeasurableSpace = generateFrom π
s : (i : ι) → Set (X i)
t : (i : κ) → Set (Y i)
hs : ∀ (i : ι), IsClosed[inst✝⁸ i] (s i)
ht : ∀ (i : κ), IsClosed[inst✝⁵ i] (t i)
hp1 : ∀ (x : (i : ι) → X i), ∏ i, (s i).indicator (fun x ↦ 1) (x i) = (univ.pi s).indicator 1 x
hp2 : ∀ (y : (j : κ) → Y j), ∏ j, (t j).indicator (fun x ↦ 1) (y j) = (univ.pi t).indicator 1 y
this :
  ∀ (p : ((i : ι) → X i) × ((j : κ) → Y j)),
    Tendsto (ofNNReal ∘ fun x ↦ (∏ c, (⋯.apprSeq x) (p.1 c)) * ∏ c, (⋯.apprSeq x) (p.2 c)) atTop
      (𝓝 ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p))
n : ℕ
ω : ((i : ι) → X i) × ((j : κ) → Y j)
⊢ ∀ i ∈ Finset.univ, (⋯.apprSeq n) (ω.1 i) ≤ 1

Tactic:
exact fun i _ ↦ HasOuterApproxClosed.apprSeq_apply_le_one (hs i) _ _

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

Goal state before:
pp: case h1
ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁵ : (j : κ) → TopologicalSpace (Y j)
inst✝⁴ : ∀ (j : κ), BorelSpace (Y j)
inst✝³ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝² : Fintype ι
inst✝¹ : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝ : IsFiniteMeasure μ
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ≥0) (g : (j : κ) → Y j →ᵇ ℝ≥0),
    ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂μ =
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂ν
hμν : μ univ = ν univ
this✝ : IsFiniteMeasure ν
π : Set (Set (((i : ι) → X i) × ((j : κ) → Y j))) :=
  image2 (fun s t ↦ s ×ˢ t) (univ.pi '' univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s})
    (univ.pi '' univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t})
hπ1 : IsPiSystem π
hπ2 : Prod.instMeasurableSpace = generateFrom π
s : (i : ι) → Set (X i)
t : (i : κ) → Set (Y i)
hs : ∀ (i : ι), IsClosed[inst✝⁸ i] (s i)
ht : ∀ (i : κ), IsClosed[inst✝⁵ i] (t i)
hp1 : ∀ (x : (i : ι) → X i), ∏ i, (s i).indicator (fun x ↦ 1) (x i) = (univ.pi s).indicator 1 x
hp2 : ∀ (y : (j : κ) → Y j), ∏ j, (t j).indicator (fun x ↦ 1) (y j) = (univ.pi t).indicator 1 y
this :
  ∀ (p : ((i : ι) → X i) × ((j : κ) → Y j)),
    Tendsto (ofNNReal ∘ fun x ↦ (∏ c, (⋯.apprSeq x) (p.1 c)) * ∏ c, (⋯.apprSeq x) (p.2 c)) atTop
      (𝓝 ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p))
h1 :
  Tendsto
    (fun n ↦
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((∏ i, (⋯.apprSeq n) (p.1 i)) * ∏ j, (⋯.apprSeq n) (p.2 j)) ∂μ)
    atTop (𝓝 (∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p) ∂μ))
x✝¹ : ℕ
x✝ : ((i : ι) → X i) × ((j : κ) → Y j)
⊢ ∀ i ∈ Finset.univ, (⋯.apprSeq x✝¹) (x✝.1 i) ≤ 1

Tactic:
exact fun i _ ↦ HasOuterApproxClosed.apprSeq_apply_le_one (hs i) _ _

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

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n).real {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
C : ℝ≥0
hμ : ∀ᶠ (n : ℕ) in atTop, (μ n) Set.univ ≤ ↑C
z : E
hz : ‖z‖ = 1
r : ℝ
⊢ limsup (fun n ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop =
    ENNReal.ofReal (limsup (fun n ↦ ((μ n) {x | r < ‖⟪z, x⟫_𝕜‖}).toReal) atTop)

Tactic:
rw [ENNReal.ofReal_limsup_toReal (C := C)]

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n).real {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
C : ℝ≥0
hμ : ∀ᶠ (n : ℕ) in atTop, (μ n) Set.univ ≤ ↑C
z : E
hz : ‖z‖ = 1
r : ℝ
⊢ ∀ᶠ (a : ℕ) in atTop, (μ a) {x | r < ‖⟪z, x⟫_𝕜‖} ≤ ↑C
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n).real {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
C : ℝ≥0
hμ : ∀ᶠ (n : ℕ) in atTop, (μ n) Set.univ ≤ ↑C
z : E
hz : ‖z‖ = 1
r : ℝ
⊢ ∀ᶠ (a : ℕ) in atTop, (μ a) {x | r < ‖⟪z, x⟫_𝕜‖} ≤ ↑C

Tactic:
filter_upwards [hμ] with n hμn using (measure_mono (Set.subset_univ _)).trans hμn

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

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
⊢ ennrealPreVariation 0 ⋯ ⋯ = 0

Tactic:
ext; simp [ennrealPreVariation_apply]

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✝ : HereditarilyLindelofSpace X
s : Set X
hμ : 0 < μ s
⊢ (s ∩ μ.support).Nonempty

Tactic:
rw [← Set.not_disjoint_iff_nonempty_inter]
  contrapose! hμ
  exact μ.mono hμ.subset_compl_right |>.trans <| by simp

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : MeasurableSpace 𝕜
inst✝⁹ : CompleteSpace 𝕜
inst✝⁸ : NormedDivisionRing A
inst✝⁷ : NormedAlgebra 𝕜 A
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : OpensMeasurableSpace 𝕜
inst✝⁴ : CompleteSpace A
inst✝³ : SecondCountableTopology A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
⊢ Integrable (resolvent a) μ

Tactic:
by
  refine ⟨by fun_prop, ?_⟩
  apply HasFiniteIntegral.of_bounded
  filter_upwards [support_mem_ae] with x hx using norm_resolvent_le_inv_infDist_support hz hx

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

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
h✝ : ¬IsFiniteMeasure μ
⊢ ae ⋯.choose = ae μ

Tactic:
· obtain ⟨_, h₁, h₂⟩ := (exists_isFiniteMeasure_absolutelyContinuous μ).choose_spec
    exact h₂.ae_le.antisymm h₁.ae_le

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_sigma_subadd : m.IsSigmaSubadditive
this : MeasurableSpace α := (inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯).caratheodory
⊢ (inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯).trim ≤ inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯

Tactic:
by
      apply le_inducedOuterMeasure.mpr fun s hs ↦ ?_
      have hs_meas : MeasurableSet[(inducedOuterMeasure (fun x _ ↦ m x) hC.empty_mem
          addContent_empty).caratheodory] s := by
        change (inducedOuterMeasure (fun x _ ↦ m x) hC.empty_mem addContent_empty).IsCaratheodory s
        exact isCaratheodory_inducedOuterMeasure_of_mem hC m hs
      rw [OuterMeasure.trim_eq _ hs_meas, m.inducedOuterMeasure_eq hC m_sigma_subadd hs]

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: case hs
α : Type u_1
C : Set (Set α)
s : Set α
mα : MeasurableSpace α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
hC_gen : mα = MeasurableSpace.generateFrom C
m_sigma_subadd : m.IsSigmaSubadditive
hs : s ∈ C
⊢ MeasurableSet s

Tactic:
hC_gen

Goal state after:
case hs
α : Type u_1
C : Set (Set α)
s : Set α
mα : MeasurableSpace α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
hC_gen : mα = MeasurableSpace.generateFrom C
m_sigma_subadd : m.IsSigmaSubadditive
hs : s ∈ C
⊢ MeasurableSet s
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case refine_2.inr
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 = ∞
⊢ ∃ t ∈ ℬ, μ (s ∆ t) < ENNReal.ofReal ε

Tactic:
refine False.elim <| hμs ?_

Goal state after:
case refine_2.inr
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 = ∞
⊢ μ s = ∞
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
h : μ.variation = 0
s : Set X
hs : MeasurableSet s
⊢ ‖μ s‖ₑ = 0

Tactic:
apply le_antisymm ?_ (by simp)

Goal state after:
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
h : μ.variation = 0
s : Set X
hs : MeasurableSet s
⊢ ‖μ s‖ₑ ≤ 0
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
hbound : ∀ (m : ℕ), ∀ μ ∈ S, 1 - ε * 2 ^ (-↑m) < ↑(μ (⋃ i, ⋃ (_ : i ≤ km m), ball (D i) (u m)))
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),
      ↑μ
        (⋃ i,
            ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m)))ᶜ =
    ∑' (m : ℕ),
      (1 -
        ↑μ
          (⋃ i,
            ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))))

Tactic:
by
      congr! with m; rw [measure_compl (by measurability) (by simp)]; simp

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.AddContent

Goal state before:
pp: α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
m : Set α → E
hm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s
inst✝ : IsFiniteMeasure μ
h'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t
h''m : ∀ (s : Set α), ¬MeasurableSet s → m s = 0
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Function.onFun Disjoint f)
⊢ Filter.Tendsto (fun e ↦ ‖m (⋃ i, f i) - ∑ i ∈ Finset.range e, m (f i)‖ₑ.toReal) Filter.atTop (nhds (ENNReal.toReal 0))

Tactic:
apply (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp

Goal state after:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
m : Set α → E
hm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s
inst✝ : IsFiniteMeasure μ
h'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t
h''m : ∀ (s : Set α), ¬MeasurableSet s → m s = 0
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Function.onFun Disjoint f)
⊢ Filter.Tendsto (fun e ↦ ‖m (⋃ i, f i) - ∑ i ∈ Finset.range e, m (f i)‖ₑ) Filter.atTop (nhds 0)
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
μ : VectorMeasure X F
s : Set X
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace ℝ F
inst✝² : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : IsFiniteMeasure (μ.variation.restrict s)
c : E
hs : MeasurableSet s
this : IsFiniteMeasure (μ.restrict s).variation
⊢ (B c) (μ (univ ∩ s)) = (B c) (μ s)

Tactic:
univ_inter

Goal state after:
case pos
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
μ : VectorMeasure X F
s : Set X
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace ℝ F
inst✝² : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : IsFiniteMeasure (μ.variation.restrict s)
c : E
hs : MeasurableSet s
this : IsFiniteMeasure (μ.restrict s).variation
⊢ (B c) (μ s) = (B c) (μ s)
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation

Goal state before:
pp: X : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
mX : MeasurableSpace X
μ : VectorMeasure X E
h : μ.semivariation univ = ∞
t : Set X → Set X
t_meas : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → MeasurableSet (t s)
t_subs : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → t s ⊆ s
t_var : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → μ.semivariation (t s) = ∞
ht : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → 1 ≤ ‖μ (s \ t s)‖ₑ
s : ℕ → Set X := fun n ↦ t^[n] univ
hs : ∀ (n : ℕ), MeasurableSet (s n) ∧ μ.semivariation (s n) = ∞
u : ℕ → Set X := fun n ↦ s n \ s (n + 1)
hu : ∀ (n : ℕ), 1 ≤ ‖μ (u n)‖ₑ
s_anti : Antitone s
⊢ False

Tactic:
have u_disj : Pairwise (Disjoint on u) := by
    apply (pairwise_disjoint_on _).2 (fun m n hmn ↦ ?_)
    have : Disjoint (u m) (s (m + 1)) := by simp [u, disjoint_sdiff_left]
    apply this.mono_right
    simp only [sdiff_le_iff, sup_eq_union, u]
    exact Subset.trans (s_anti (by grind)) subset_union_right

Goal state after:
X : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
mX : MeasurableSpace X
μ : VectorMeasure X E
h : μ.semivariation univ = ∞
t : Set X → Set X
t_meas : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → MeasurableSet (t s)
t_subs : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → t s ⊆ s
t_var : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → μ.semivariation (t s) = ∞
ht : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → 1 ≤ ‖μ (s \ t s)‖ₑ
s : ℕ → Set X := fun n ↦ t^[n] univ
hs : ∀ (n : ℕ), MeasurableSet (s n) ∧ μ.semivariation (s n) = ∞
u : ℕ → Set X := fun n ↦ s n \ s (n + 1)
hu : ∀ (n : ℕ), 1 ≤ ‖μ (u n)‖ₑ
s_anti : Antitone s
u_disj : Pairwise (Disjoint on u)
⊢ False
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
s : Set M
F : Type u_6
inst✝¹ : EquivLike F M N
inst✝ : AddEquivClass F M N
f : F
⊢ IsSemilinearSet (⇑f '' s) ↔ IsSemilinearSet s

Tactic:
constructor <;> intro h

Goal state after:
case mp
M : Type u_1
N : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
s : Set M
F : Type u_6
inst✝¹ : EquivLike F M N
inst✝ : AddEquivClass F M N
f : F
h : IsSemilinearSet (⇑f '' s)
⊢ IsSemilinearSet s

case mpr
M : Type u_1
N : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
s : Set M
F : Type u_6
inst✝¹ : EquivLike F M N
inst✝ : AddEquivClass F M N
f : F
h : IsSemilinearSet s
⊢ IsSemilinearSet (⇑f '' s)
Module: Mathlib.MeasureTheory.VectorMeasure.Integral

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : Nontrivial E
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = ‖x‖₊ * ‖y‖₊
this : μ.variation = 1 • μ.variation
⊢ ∀ (x : E) (y : F), ‖(B x) y‖₊ = 1 * ‖x‖₊ * ‖y‖₊

Tactic:
simpa using hB

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: case neg
X : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
f : X → E
B : E →L[ℝ] F →L[ℝ] G
hf : ¬μ.Integrable f
⊢ (μ.withDensity f B) univ = ∫ᵛ (x : X), f x ∂[B; μ]

Tactic:
simp [withDensity, hf, integral_undef]

Goal state after: