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)
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)
⊢ volume {x | ∑ i, |x i| ^ p < 1} = ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))

Tactic:
norm_eq_abs

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)
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)
⊢ 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: case h₁
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
C : ℝ≥0
K : Set E
hK : IsCompact K
f : ↑K → E := Subtype.val
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f
T : Set (FiniteMeasure ↑K) := {μ | μ.mass ≤ C}
μ : FiniteMeasure E
hμ : μ ∈ {μ | μ.mass ≤ C ∧ μ Kᶜ = 0}
s : Set E
hs : MeasurableSet s
⊢ (Measure.map f ↑(FiniteMeasure.comap f μ)) s = ↑μ s

Tactic:
rw [Measure.map_apply measurable_subtype_coe hs]

Goal state after:
case h₁
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
C : ℝ≥0
K : Set E
hK : IsCompact K
f : ↑K → E := Subtype.val
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f
T : Set (FiniteMeasure ↑K) := {μ | μ.mass ≤ C}
μ : FiniteMeasure E
hμ : μ ∈ {μ | μ.mass ≤ C ∧ μ Kᶜ = 0}
s : Set E
hs : MeasurableSet s
⊢ ↑(FiniteMeasure.comap f μ) (Subtype.val ⁻¹' s) = ↑μ s
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
⊢ 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_neg := fun x : ι → ℝ => norm_neg (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‖
⊢ 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.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)
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)
⊢ 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:
norm_eq_abs

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)
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)
⊢ 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.Lebesgue.VolumeOfBalls

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
⊢ 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:
by
  have h₁ : 0 < p := by linarith
  have : (ENNReal.ofReal p).toReal = p := toReal_ofReal (le_of_lt h₁)
  -- We collect facts about `Lp` norms that will be used in `measure_le_one_eq_lt_one`
  have eq_norm (x : ι → ℝ) : ‖toLp (.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p) := by
    simp [PiLp.norm_eq_sum (f := toLp (.ofReal p) x) (this.symm ▸ h₁), this]
  have : Fact (1 ≤ ENNReal.ofReal p) := fact_iff.mpr (ofReal_one ▸ (ofReal_le_ofReal hp))
  have nm_zero : ‖toLp (.ofReal p) (0 : ι → ℝ)‖ = 0 := norm_zero
  have eq_zero (x : ι → ℝ) : ‖toLp (.ofReal p) x‖ = 0 ↔ x = 0 :=
    norm_eq_zero.trans (toLp_eq_zero _)
  have nm_neg := fun x : ι → ℝ => norm_neg (toLp (.ofReal p) x)
  have nm_add := fun x y : ι → ℝ => norm_add_le (toLp (.ofReal p) x) (toLp (.ofReal p) y)
  have nm_smul := fun (r : ℝ) (x : ι → ℝ) =>
    norm_smul_le (β := PiLp (.ofReal p) (fun _ : ι => ℝ)) r (toLp (.ofReal p) x)
  simp_rw [← toLp_neg, ← toLp_add, ← toLp_smul, eq_norm,
    norm_eq_abs] at eq_zero nm_zero nm_neg nm_add nm_smul
  rw [measure_le_eq_lt _ nm_zero (fun x ↦ nm_neg x) (fun x y ↦ nm_add x y) (eq_zero _).mp
    (fun r x => nm_smul r x), volume_sum_rpow_lt _ hp]

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 → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℂ), (∑ i, ‖x i‖ ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ i, ‖0 i‖ ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℂ), (∑ i, ‖(-x) i‖ ^ p) ^ (1 / p) = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
nm_add : ∀ (x y : ι → ℂ), (∑ 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 → ℂ), (∑ i, ‖(r • x) i‖ ^ p) ^ (1 / p) ≤ ‖r‖ * (∑ i, ‖x i‖ ^ p) ^ (1 / p)
⊢ 2 * ↑(card ι) / p + 1 = ↑Finset.univ.card * 2 / p + 1

Tactic:
Fintype.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 → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℂ), (∑ i, ‖x i‖ ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ i, ‖0 i‖ ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℂ), (∑ i, ‖(-x) i‖ ^ p) ^ (1 / p) = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
nm_add : ∀ (x y : ι → ℂ), (∑ 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 → ℂ), (∑ i, ‖(r • x) i‖ ^ p) ^ (1 / p) ≤ ‖r‖ * (∑ i, ‖x i‖ ^ p) ^ (1 / p)
⊢ 2 * ↑Finset.univ.card / p + 1 = ↑Finset.univ.card * 2 / p + 1
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd

Goal state before:
pp: case hx
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✝⁹ : (i : ι) → TopologicalSpace (X i)
inst✝⁸ : ∀ (i : ι), BorelSpace (X i)
inst✝⁷ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁶ : (j : κ) → TopologicalSpace (Y j)
inst✝⁵ : ∀ (j : κ), BorelSpace (Y j)
inst✝⁴ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝³ : Fintype ι
inst✝² : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ) (g : (j : κ) → Y j →ᵇ ℝ),
    ∫ (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
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 ≠ ∞

Tactic:
convert!
    (lintegral_lt_top_of_nnreal μ
        ((∏ i, (f i).compContinuous ⟨Function.eval i ∘ Prod.fst, by fun_prop⟩) *
          (∏ j, (g j).compContinuous ⟨Function.eval j ∘ Prod.snd, by fun_prop⟩))).ne
    simp

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
left✝ : IsFiniteMeasure ⋯.choose
h₁ : μ ≪ ⋯.choose
h₂ : ⋯.choose ≪ μ
⊢ ae ⋯.choose = ae μ

Tactic:
exact h₂.ae_le.antisymm h₁.ae_le

Goal state after:
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:
by
  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.Prokhorov

Goal state before:
pp: case hK
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
S : Set (ProbabilityMeasure E)
hS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}
u : ℕ → ℝ≥0
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
K : ℕ → Set E
K_comp : ∀ (n : ℕ), IsCompact (K n)
hK : ∀ (n : ℕ), ∀ μ ∈ S, μ (K n)ᶜ ≤ u n
K' : ℕ → Set E := ⋯
⊢ ∀ (n : ℕ), IsCompact (K' n)

pp: case h
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
S : Set (ProbabilityMeasure E)
hS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}
u : ℕ → ℝ≥0
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
K : ℕ → Set E
K_comp : ∀ (n : ℕ), IsCompact (K n)
hK : ∀ (n : ℕ), ∀ μ ∈ S, μ (K n)ᶜ ≤ u n
K' : ℕ → Set E := ⋯
⊢ NormalSpace E ∨ Monotone K'

Tactic:
· exact fun n ↦ (finite_Iic n).isCompact_biUnion (fun i hi ↦ K_comp i)

Goal state after:
case h
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
S : Set (ProbabilityMeasure E)
hS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}
u : ℕ → ℝ≥0
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
K : ℕ → Set E
K_comp : ∀ (n : ℕ), IsCompact (K n)
hK : ∀ (n : ℕ), ∀ μ ∈ S, μ (K n)ᶜ ≤ u n
K' : ℕ → Set E := ⋯
⊢ NormalSpace E ∨ Monotone K'
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).snd = ∫ (x : X), (f x).snd ∂μ

Tactic:
rw [← WithLp.ofLp_snd]

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
⊢ (∫ (x : X), f x ∂μ).ofLp.2 = ∫ (x : X), (f x).snd ∂μ
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
s : Set X
hs : MeasurableSet s
P : Finset (Set X)
hP₁ : ∀ t ∈ P, t ⊆ s
hP₂ : (↑P).PairwiseDisjoint id
Q : Finpartition (P.sup id) := ⋯
defQ : Q = Finpartition.ofPairwiseDisjoint P hP₂
Q' : Finpartition ((filter MeasurableSet Q.parts).sup id) := ⋯
defQ' : Q' = Q.ofSubset ⋯ ⋯
hQ' : ∀ t ∈ Q'.parts, t ⊆ s
⊢ ∀ p ∈ (Q'.extendOfLE ⋯).parts, MeasurableSet p

Tactic:
intro p hp

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
s : Set X
hs : MeasurableSet s
P : Finset (Set X)
hP₁ : ∀ t ∈ P, t ⊆ s
hP₂ : (↑P).PairwiseDisjoint id
Q : Finpartition (P.sup id) := Finpartition.ofPairwiseDisjoint P hP₂
defQ : Q = Finpartition.ofPairwiseDisjoint P hP₂
Q' : Finpartition ((filter MeasurableSet Q.parts).sup id) := Q.ofSubset ⋯ ⋯
defQ' : Q' = Q.ofSubset ⋯ ⋯
hQ' : ∀ t ∈ Q'.parts, t ⊆ s
p : Set X
hp : p ∈ (Q'.extendOfLE ⋯).parts
⊢ MeasurableSet p
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: case refine_1
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
s : Set X
hs : MeasurableSet s
a : ℝ≥0∞
ha : a < preVariationFun (fun x ↦ ‖μ x‖ₑ) s
P : Finpartition ⟨s, hs⟩
hP : a < ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p
t : Set X
⊢ ∀ (x : MeasurableSet t), ⟨t, ⋯⟩ ∈ P.parts → t ⊆ s

Tactic:
ht

Goal state after:
case refine_1
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
s : Set X
hs : MeasurableSet s
a : ℝ≥0∞
ha : a < preVariationFun (fun x ↦ ‖μ x‖ₑ) s
P : Finpartition ⟨s, hs⟩
hP : a < ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p
t : Set X
ht : MeasurableSet t
⊢ ⟨t, ⋯⟩ ∈ P.parts → t ⊆ s
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: case refine_1
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
s : Set X
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
hμ : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≠ ∞
P : Finpartition ⟨s, hs⟩
hP : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≤ ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p + ε
t : Set X
⊢ ∀ (x : MeasurableSet t), ⟨t, ⋯⟩ ∈ P.parts → t ⊆ s

Tactic:
ht

Goal state after:
case refine_1
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
s : Set X
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
hμ : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≠ ∞
P : Finpartition ⟨s, hs⟩
hP : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≤ ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p + ε
t : Set X
ht : MeasurableSet t
⊢ ⟨t, ⋯⟩ ∈ P.parts → t ⊆ s
Module: Mathlib.MeasureTheory.VectorMeasure.AddContent

Goal state before:
pp: case a
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)
A : ∀ (n : ℕ), m (⋃ i ∈ Finset.range n, f i) = ∑ i ∈ Finset.range n, m (f i)
n : ℕ
this : ⋃ i, f i = (⋃ i ∈ Finset.range n, f i) ∪ ⋃ i ∈ Set.Ici n, f i
⊢ MeasurableSet (⋃ i ∈ Set.Ici n, f i)

Tactic:
exact MeasurableSet.biUnion (Set.to_countable _) (fun i hi ↦ f_meas i)

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

Goal state before:
pp: case a
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)
A : ∀ (n : ℕ), m (⋃ i ∈ Finset.range n, f i) = ∑ i ∈ Finset.range n, m (f i)
n : ℕ
this : ⋃ i, f i = (⋃ i ∈ Finset.range n, f i) ∪ ⋃ i ∈ Set.Ici n, f i
⊢ Disjoint (⋃ i ∈ Finset.range n, f i) (⋃ i ∈ Set.Ici n, f i)

Tactic:
simp only [Finset.mem_range, Set.mem_Ici, Set.disjoint_iUnion_right,
          Set.disjoint_iUnion_left]
        intro i hi j hj
        exact f_disj (hj.trans_le hi).ne

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

Goal state before:
pp: case inr
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
a b : α
hf : BoundedVariationOn f univ
h : a ≤ b
h' : a < b
⊢ hf.vectorMeasure (Ico a b) = Function.leftLim f b - Function.leftLim f a

Tactic:
rw [← Icc_union_Ioo_eq_Ico le_rfl h', VectorMeasure.of_union (by simp) measurableSet_Icc
    measurableSet_Ioo, hf.vectorMeasure_Icc le_rfl, hf.vectorMeasure_Ioo h']

Goal state after:
case inr
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
a b : α
hf : BoundedVariationOn f univ
h : a ≤ b
h' : a < b
⊢ Function.rightLim f a - Function.leftLim f a + (Function.leftLim f b - Function.rightLim f a) =
    Function.leftLim f b - 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 atTop (𝓝 (atTop.limUnder f))
u : ℕ → α
u_mono : Monotone u
hu : Tendsto u atTop atTop
⊢ ⋃ n, Icc a (u n) ⊆ Ici a

Tactic:
by simp [Icc_subset_Ici_self]

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

Goal state before:
pp: X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
t : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
hU : ∫ᵛ (x : X) in t, f x ∂[B; μ] ≠ 0
ht : MeasurableSet t
⊢ ∃ᵐ (x : X) ∂μ.variation.restrict t, f x ≠ 0

Tactic:
rw [← variation_restrict ht]

Goal state after:
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
t : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
hU : ∫ᵛ (x : X) in t, f x ∂[B; μ] ≠ 0
ht : MeasurableSet t
⊢ ∃ᵐ (x : X) ∂(μ.restrict t).variation, f x ≠ 0
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 : (fun n ↦ Function.rightLim f a - Function.leftLim f (u n)) =ᶠ[atTop] fun n ↦ hf.vectorMeasure (Icc (u n) a)
⊢ Tendsto (fun n ↦ Function.rightLim f a - Function.leftLim f (u n)) atTop
    (𝓝 (Function.rightLim f a - atBot.limUnder f))

Tactic:
apply Tendsto.sub tendsto_const_nhds

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 : (fun n ↦ Function.rightLim f a - Function.leftLim f (u n)) =ᶠ[atTop] fun n ↦ hf.vectorMeasure (Icc (u n) a)
⊢ Tendsto (fun x ↦ Function.leftLim f (u x)) atTop (𝓝 (atBot.limUnder f))