mathlib-initiative/mathlib-tactics huggingface dataset:
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
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
h_limsup_le :
  ∀ (r : ℝ),
    0 < r →
      limsup (fun n ↦ (μ n).real {x | r < |⟪z, x⟫|}) atTop ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)‖
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ r
hr : 0 < r
⊢ -(2 * r⁻¹) ≤ 2 * r⁻¹

Tactic:
neg_le_self_iff

Goal state after:
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
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
h_limsup_le :
  ∀ (r : ℝ),
    0 < r →
      limsup (fun n ↦ (μ n).real {x | r < |⟪z, x⟫|}) atTop ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)‖
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ r
hr : 0 < r
⊢ 0 ≤ 2 * r⁻¹
Module: Mathlib.MeasureTheory.Measure.PreVariation

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

Tactic:
have hε : 0 < ε := by positivity

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

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
hf : IsSigmaSubadditiveSetFun f
hf' : f ∅ = 0
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
b : ℝ≥0∞
Q : Finpartition ⟨⋃ i, s i, ⋯⟩
hQ : b < ∑ p ∈ Q.parts, f ↑p
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
P : (i : ℕ) → Finpartition (s' i) := fun i ↦ Q.restrict ⋯
⊢ ∑ q ∈ Q.parts, f ↑q ≤ ∑ q ∈ Q.parts, ∑' (i : ℕ), f (↑q ⊓ ↑(s' i))

Tactic:
by
          apply Finset.sum_le_sum fun q hq => ?_
          have hq_eq : q.val = ⋃ i, q.val ∩ s i := by
            rw [← Set.inter_iUnion]; exact (Set.inter_eq_left.mpr (Q.le hq)).symm
          let t (i : ℕ) : Subtype MeasurableSet := ⟨q.val ∩ s i, q.2.inter (hs i)⟩
          have ht_disj : Pairwise (Disjoint on (Subtype.val ∘ t)) :=
            fun i j hij => (hs' hij).mono Set.inter_subset_right Set.inter_subset_right
          calc f q
            _ = f (⋃ i, q.val ∩ s i) := congrArg f hq_eq
            _ = f (⋃ i, (t i).val) := rfl
            _ ≤ ∑' i, f (t i) := hf t ht_disj
            _ = ∑' i, f (q ⊓ s' i) := rfl

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

Goal state before:
pp: case inl
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
⊢ {x | ∃ s, ∃ (hs : MeasurableSet s) (hμs : μ s ≠ ∞), indicatorConstLp p hs hμs 0 = x} ⊆
    closure {x | ∃ s, ∃ (hs : s ∈ 𝒜) (hμs : μ s ≠ ∞), indicatorConstLp p ⋯ hμs 0 = x}

Tactic:
refine Subset.trans ?_ subset_closure

Goal state after:
case inl
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
⊢ {x | ∃ s, ∃ (hs : MeasurableSet s) (hμs : μ s ≠ ∞), indicatorConstLp p hs hμs 0 = x} ⊆
    {x | ∃ s, ∃ (hs : s ∈ 𝒜) (hμs : μ s ≠ ∞), indicatorConstLp p ⋯ hμs 0 = x}
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case inl
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
⊢ {x | ∃ s, ∃ (hs : MeasurableSet s) (hμs : μ s ≠ ∞), indicatorConstLp p hs hμs 0 = x} ⊆
    closure {x | ∃ s, ∃ (hs : s ∈ 𝒜) (hμs : μ s ≠ ∞), indicatorConstLp p ⋯ hμs 0 = x}

pp: case inr
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
c : E
hc : c ≠ 0
⊢ {x | ∃ s, ∃ (hs : MeasurableSet s) (hμs : μ s ≠ ∞), indicatorConstLp p hs hμs c = x} ⊆
    closure {x | ∃ s, ∃ (hs : s ∈ 𝒜) (hμs : μ s ≠ ∞), indicatorConstLp p ⋯ hμs c = x}

Tactic:
· refine Subset.trans ?_ subset_closure
    rintro - ⟨s, ms, hμs, rfl⟩
    obtain ⟨t, ht, hμt⟩ := h𝒜.nonempty'
    refine ⟨t, ht, hμt, ?_⟩
    simp_rw [indicatorConstLp]
    simp

Goal state after:
case inr
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
c : E
hc : c ≠ 0
⊢ {x | ∃ s, ∃ (hs : MeasurableSet s) (hμs : μ s ≠ ∞), indicatorConstLp p hs hμs c = x} ⊆
    closure {x | ∃ s, ∃ (hs : s ∈ 𝒜) (hμs : μ s ≠ ∞), indicatorConstLp p ⋯ hμs c = x}
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : PolishSpace E
inst✝ : BorelSpace E
ι : Type u_4
𝓕 : Filter ι
μ : ι → ProbabilityMeasure E
h_tight : IsTightMeasureSet {x | ∃ n, ↑(μ n) = x}
μ₀ : ProbabilityMeasure E
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
hμ : ∀ g ∈ A, Tendsto (fun n ↦ ∫ (x : E), g x ∂↑(μ n)) 𝓕 (𝓝 (∫ (x : E), g x ∂↑μ₀))
⊢ Tendsto μ 𝓕 (𝓝 μ₀)

Tactic:
let := TopologicalSpace.upgradeIsCompletelyMetrizable E
  obtain rfl | _ := 𝓕.eq_or_neBot
  · simp
  refine (Filter.tendsto_iff_ultrafilter _ _ _).2 fun U hU ↦ ?_
  have h_compact : IsCompact (closure {μ n | n}) :=
    isCompact_closure_of_isTightMeasureSet (by simpa using h_tight)
  obtain ⟨μ', -, hμ' : Tendsto _ _ _⟩ := h_compact.ultrafilter_le_nhds (U.map μ)
    (.trans (by simp) (monotone_principal subset_closure))
  suffices (μ' : Measure E) = μ₀ by convert! hμ'; ext; rw [this]
  refine ext_of_forall_mem_subalgebra_integral_eq_of_pseudoEMetric_complete_countable hA
    fun g hg ↦ tendsto_nhds_unique ?_ ((hμ g hg).comp hU)
  rw [ProbabilityMeasure.tendsto_iff_forall_integral_rclike_tendsto 𝕜] at hμ'
  exact hμ' g

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

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

Tactic:
by
                          simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul,
                            Nat.cast_add, Nat.cast_one]
                          rw [div_mul_eq_div_mul_one_div, ← mul_assoc, mul_comm ((N : ℝ) + 1),
                            mul_assoc]
                          exact mul_le_of_le_one_right (by linarith [ε_pos]) <|
                            le_of_eq <| mul_one_div_cancel <| Nat.cast_add_one_ne_zero _

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

Goal state before:
pp: case compl
α : Type u_1
C : Set (Set α)
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
s t : Set α
a✝ : MeasurableSpace.GenerateMeasurable C t
h : (inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯).IsCaratheodory t
⊢ (inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯).IsCaratheodory tᶜ

Tactic:
| compl t _ h =>

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

Goal state before:
pp: case insert
ι : Type u_1
X : Type u_2
E : Type u_3
F : Type u_4
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
μ : VectorMeasure X F
f : X → E
s : Set ι
t : ι → Set X
a✝¹ : ι
s✝ : Set ι
a✝ : a✝¹ ∉ s✝
h's : s✝.Finite
hf : (∀ i ∈ s✝, MeasurableSet (t i)) → (∀ i ∈ s✝, μ.IntegrableOn f (t i)) → μ.IntegrableOn f (⋃ i ∈ s✝, t i)
ht : MeasurableSet (t a✝¹) ∧ ∀ a ∈ s✝, MeasurableSet (t a)
h't : μ.IntegrableOn f (t a✝¹) ∧ ∀ a ∈ s✝, μ.IntegrableOn f (t a)
⊢ μ.IntegrableOn f (t a✝¹ ∪ ⋃ x ∈ s✝, t x)

Tactic:
exact IntegrableOn.union ht.1 (h's.measurableSet_biUnion ht.2)  h't.1 (hf ht.2 h't.2)

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

Goal state before:
pp: case h_add
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedAddCommGroup H
μ : VectorMeasure X F
f✝ : X → E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace H
C : G →L[ℝ] H
hf : Integrable f✝ μ.variation
f : X → E
⊢ ∀ ⦃g : X → E⦄,
    Disjoint (support f) (support g) →
      Integrable f μ.variation →
        Integrable g μ.variation →
          C ∫ᵛ (y : X), f y ∂[B; μ] = ∫ᵛ (y : X), f y ∂[(compL ℝ F G H) C ∘SL B; μ] →
            C ∫ᵛ (y : X), g y ∂[B; μ] = ∫ᵛ (y : X), g y ∂[(compL ℝ F G H) C ∘SL B; μ] →
              C ∫ᵛ (y : X), (f + g) y ∂[B; μ] = ∫ᵛ (y : X), (f + g) y ∂[(compL ℝ F G H) C ∘SL B; μ]

Tactic:
g

Goal state after:
case h_add
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedAddCommGroup H
μ : VectorMeasure X F
f✝ : X → E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace H
C : G →L[ℝ] H
hf : Integrable f✝ μ.variation
f g : X → E
⊢ Disjoint (support f) (support g) →
    Integrable f μ.variation →
      Integrable g μ.variation →
        C ∫ᵛ (y : X), f y ∂[B; μ] = ∫ᵛ (y : X), f y ∂[(compL ℝ F G H) C ∘SL B; μ] →
          C ∫ᵛ (y : X), g y ∂[B; μ] = ∫ᵛ (y : X), g y ∂[(compL ℝ F G H) C ∘SL B; μ] →
            C ∫ᵛ (y : X), (f + g) y ∂[B; μ] = ∫ᵛ (y : X), (f + g) y ∂[(compL ℝ F G H) C ∘SL B; μ]
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
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ supClosure C, μ (s ∆ t) < ε
m₀ : AddContent E (supClosure C) := m.supClosure hC
hm₀ : m₀ = m.supClosure hC
s : Set α
hs : ∃ P, ↑P.parts ⊆ C
⊢ ‖m₀ s‖ₑ ≤ μ s

Tactic:
rcases hs with ⟨P, PC⟩

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
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ supClosure C, μ (s ∆ t) < ε
m₀ : AddContent E (supClosure C) := m.supClosure hC
hm₀ : m₀ = m.supClosure hC
s : Set α
P : Finpartition s
PC : ↑P.parts ⊆ C
⊢ ‖m₀ s‖ₑ ≤ μ s
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
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ supClosure C, μ (s ∆ t) < ε
m₀ : AddContent E (supClosure C) := m.supClosure hC
hm₀ : m₀ = m.supClosure hC
A : ∀ s ∈ supClosure C, ‖m₀ s‖ₑ ≤ μ s
s : Set α
hs : ∃ P, ↑P.parts ⊆ C
⊢ MeasurableSet s

Tactic:
rcases hs with ⟨P, PC⟩

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
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ supClosure C, μ (s ∆ t) < ε
m₀ : AddContent E (supClosure C) := m.supClosure hC
hm₀ : m₀ = m.supClosure hC
A : ∀ s ∈ supClosure C, ‖m₀ s‖ₑ ≤ μ s
s : Set α
P : Finpartition s
PC : ↑P.parts ⊆ C
⊢ MeasurableSet s
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case h_ae
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedAddCommGroup H
μ : VectorMeasure X F
f✝ : X → E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace H
C : G →L[ℝ] H
hf : Integrable f✝ μ.variation
f : X → E
⊢ ∀ ⦃g : X → E⦄,
    f =ᵐ[μ.variation] g →
      Integrable f μ.variation →
        C ∫ᵛ (y : X), f y ∂[B; μ] = ∫ᵛ (y : X), f y ∂[(compL ℝ F G H) C ∘SL B; μ] →
          C ∫ᵛ (y : X), g y ∂[B; μ] = ∫ᵛ (y : X), g y ∂[(compL ℝ F G H) C ∘SL B; μ]

Tactic:
g

Goal state after:
case h_ae
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedAddCommGroup H
μ : VectorMeasure X F
f✝ : X → E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace H
C : G →L[ℝ] H
hf : Integrable f✝ μ.variation
f g : X → E
⊢ f =ᵐ[μ.variation] g →
    Integrable f μ.variation →
      C ∫ᵛ (y : X), f y ∂[B; μ] = ∫ᵛ (y : X), f y ∂[(compL ℝ F G H) C ∘SL B; μ] →
        C ∫ᵛ (y : X), g y ∂[B; μ] = ∫ᵛ (y : X), g y ∂[(compL ℝ F G H) C ∘SL B; μ]
Module: Mathlib.MeasureTheory.VectorMeasure.Prod

Goal state before:
pp: ι : Type u_1
X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
H : Type u_7
I : Type u_8
J : Type u_9
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : NormedAddCommGroup H
inst✝⁵ : NormedSpace ℝ H
inst✝⁴ : NormedAddCommGroup I
inst✝³ : NormedSpace ℝ I
inst✝² : NormedAddCommGroup J
inst✝¹ : NormedSpace ℝ J
μ✝ : VectorMeasure X E
ν✝ : VectorMeasure Y F
B✝ : E →L[ℝ] F →L[ℝ] G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : IsFiniteMeasure μ.variation
f : ℕ → Set (X × Y)
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Disjoint on f)
⊢ Tendsto (fun s ↦ ∑ x ∈ s, ∫ᵛ (x_1 : X), ν (Prod.mk x_1 ⁻¹' f x) ∂[B.flip; μ]) atTop
    (𝓝 ∫ᵛ (x : X), ν (⋃ i, Prod.mk x ⁻¹' f i) ∂[B.flip; μ])

Tactic:
have A (a : Finset ℕ) : ∑ y ∈ a, ∫ᵛ x, ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ]
        = ∫ᵛ x, ∑ y ∈ a, ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ] := by
      rw [integral_finsetSum _ (fun i hi ↦ integrable_vectorMeasure_prodMk_left (f_meas i))]

Goal state after:
ι : Type u_1
X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
H : Type u_7
I : Type u_8
J : Type u_9
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : NormedAddCommGroup H
inst✝⁵ : NormedSpace ℝ H
inst✝⁴ : NormedAddCommGroup I
inst✝³ : NormedSpace ℝ I
inst✝² : NormedAddCommGroup J
inst✝¹ : NormedSpace ℝ J
μ✝ : VectorMeasure X E
ν✝ : VectorMeasure Y F
B✝ : E →L[ℝ] F →L[ℝ] G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : IsFiniteMeasure μ.variation
f : ℕ → Set (X × Y)
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Disjoint on f)
A :
  ∀ (a : Finset ℕ),
    ∑ y ∈ a, ∫ᵛ (x : X), ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ] = ∫ᵛ (x : X), ∑ y ∈ a, ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ]
⊢ Tendsto (fun s ↦ ∑ x ∈ s, ∫ᵛ (x_1 : X), ν (Prod.mk x_1 ⁻¹' f x) ∂[B.flip; μ]) atTop
    (𝓝 ∫ᵛ (x : X), ν (⋃ i, Prod.mk x ⁻¹' f i) ∂[B.flip; μ])
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos.h_add
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup H
μ : VectorMeasure X F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
f✝ : X → H
C : H →L[ℝ] E
hf : Integrable f✝ μ.variation
hG : CompleteSpace G
f : X → H
⊢ ∀ ⦃g : X → H⦄,
    Disjoint (support f) (support g) →
      Integrable f μ.variation →
        Integrable g μ.variation →
          ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ] →
            ∫ᵛ (y : X), C (g y) ∂[B; μ] = ∫ᵛ (y : X), g y ∂[B ∘SL C; μ] →
              ∫ᵛ (y : X), C ((f + g) y) ∂[B; μ] = ∫ᵛ (y : X), (f + g) y ∂[B ∘SL C; μ]

Tactic:
g

Goal state after:
case pos.h_add
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup H
μ : VectorMeasure X F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
f✝ : X → H
C : H →L[ℝ] E
hf : Integrable f✝ μ.variation
hG : CompleteSpace G
f g : X → H
⊢ Disjoint (support f) (support g) →
    Integrable f μ.variation →
      Integrable g μ.variation →
        ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ] →
          ∫ᵛ (y : X), C (g y) ∂[B; μ] = ∫ᵛ (y : X), g y ∂[B ∘SL C; μ] →
            ∫ᵛ (y : X), C ((f + g) y) ∂[B; μ] = ∫ᵛ (y : X), (f + g) y ∂[B ∘SL C; μ]
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos.h_ae
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup H
μ : VectorMeasure X F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
f✝ : X → H
C : H →L[ℝ] E
hf : Integrable f✝ μ.variation
hG : CompleteSpace G
f : X → H
⊢ ∀ ⦃g : X → H⦄,
    f =ᵐ[μ.variation] g →
      Integrable f μ.variation →
        ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ] →
          ∫ᵛ (y : X), C (g y) ∂[B; μ] = ∫ᵛ (y : X), g y ∂[B ∘SL C; μ]

Tactic:
g

Goal state after:
case pos.h_ae
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup H
μ : VectorMeasure X F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
f✝ : X → H
C : H →L[ℝ] E
hf : Integrable f✝ μ.variation
hG : CompleteSpace G
f g : X → H
⊢ f =ᵐ[μ.variation] g →
    Integrable f μ.variation →
      ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ] →
        ∫ᵛ (y : X), C (g y) ∂[B; μ] = ∫ᵛ (y : X), g y ∂[B ∘SL C; μ]
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.SignedMeasure

Goal state before:
pp: case pos
X : Type u_1
mX : MeasurableSpace X
s : SignedMeasure X
i : Set X
hi : MeasurableSet i
⊢ ‖s i‖ ≤ s.totalVariation.real i

Tactic:
rw [s.apply_eq_posPart_real_sub_negPart_real hi, totalVariation, measureReal_add_apply]
    grind [measureReal_nonneg, Real.norm_eq_abs]

Goal state after:
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: M : Type u_1
inst✝ : AddCommMonoid M
S : Set (Set M)
hS : S.Finite
hS' : ∀ s ∈ S, IsSemilinearSet s
⊢ IsSemilinearSet (⋃₀ S)

Tactic:
by
  induction S, hS using Finite.induction_on with
  | empty => simp
  | insert _ _ ih =>
    simp_rw [mem_insert_iff, forall_eq_or_imp] at hS'
    simpa using hS'.1.union (ih hS'.2)

Goal state after:
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: M : Type u_1
inst✝ : AddCommMonoid M
S : Set (Set M)
hS : S.Finite
hS' : ∀ s ∈ S, IsProperSemilinearSet s
⊢ IsProperSemilinearSet (⋃₀ S)

Tactic:
by
  induction S, hS using Finite.induction_on with
  | empty => simp
  | insert _ _ ih =>
    simp_rw [mem_insert_iff, forall_eq_or_imp] at hS'
    simpa using hS'.1.union (ih hS'.2)

Goal state after: