mathlib-initiative/mathlib-tactics huggingface dataset:
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'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
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, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0 ∧ D.Nonempty

Tactic:
by
        rcases h'C with ⟨D, D_count, DC, hD⟩
        refine ⟨D ∪ {∅}, D_count.union (by simp), ?_⟩
        simp only [union_subset_iff, DC, singleton_subset_iff, true_and, and_true, hC.empty_mem]
        simp only [union_singleton, sUnion_insert, empty_union, insert_nonempty, and_true, hD]

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
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
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
x : ℝ
hx : x ∈ Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)
⊢ 2 * r⁻¹ < δ

Tactic:
by
      rw [← lt_div_iff₀' (by positivity), inv_lt_comm₀ hr (by positivity)]
      refine lt_of_lt_of_le ?_ hrδ
      field_simp
      norm_num

Goal state after:
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 ≤ ∑' (i : ℕ), ∑ p ∈ (P i).parts, f ↑p

Tactic:
calc ∑ q ∈ Q.parts, f q
      _ ≤ ∑ q ∈ Q.parts, ∑' i, f (q ⊓ s' i) := 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
      _ = ∑' i, ∑ q ∈ Q.parts, f (q ⊓ s' i) :=
          (Summable.tsum_finsetSum (fun _ _ ↦ ENNReal.summable)).symm
      _ = ∑' i, ∑ p ∈ (P i).parts, f p := by
          congr 1; funext i
          exact (Q.sum_restrict _ (fun p => f p) hf').symm

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✝ : OpensMeasurableSpace X
s : Set X
x : X
⊢ (∀ (i : Set X), x ∈ i ∧ IsOpen[inst✝²] i → 0 < (μ.restrict s) i) ↔
    ∀ (i : Set X), x ∈ i ∧ IsOpen[inst✝²] i → 0 < μ (i ∩ s)

Tactic:
grind [IsOpen.measurableSet, restrict_apply]

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : MeasurableSpace 𝕜
a : A
inst✝⁵ : OpensMeasurableSpace 𝕜
inst✝⁴ : NormedRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : MeasurableSpace A
inst✝ : BorelSpace A
h1 : ContinuousOn (resolvent a) (resolventSet 𝕜 a)
h2 : ContinuousOn (resolvent a) (resolventSet 𝕜 a)ᶜ
h3 : MeasurableSet (resolventSet 𝕜 a)
⊢ Measurable (resolvent a)

Tactic:
simpa using h1.measurable_piecewise h2 h3

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

Goal state before:
pp: case hab
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
x : 𝕜
hx : x ∈ μ.support
w : A
hw : w ∈ s
⊢ dist a ((algebraMap 𝕜 A) x) ≤ ‖(algebraMap 𝕜 A) x - a‖

Tactic:
dist_comm,

Goal state after:
case hab
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
x : 𝕜
hx : x ∈ μ.support
w : A
hw : w ∈ s
⊢ dist ((algebraMap 𝕜 A) x) a ≤ ‖(algebraMap 𝕜 A) x - a‖
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case hf
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
ι : Type u_2
𝓕 : Filter ι
μ₀ : ProbabilityMeasure E
μ✝ : ι → ProbabilityMeasure E
h : ∀ (t : E), Tendsto (fun n ↦ charFun (↑(μ✝ n)) t) 𝓕 (𝓝 (charFun (↑μ₀) t))
g : E →ᵇ ℂ
w : AddMonoidAlgebra ℂ E
hw : ⇑g = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(probChar (((innerₗ E) x) a))
μ : Measure E
hμ : IsProbabilityMeasure μ
⊢ ∀ i ∈ w.coeff.support, Integrable (fun x ↦ w.coeff i * ↑(probChar (((innerₗ E) x) i))) μ

Tactic:
refine fun i hi ↦ Integrable.const_mul ?_ _

Goal state after:
case hf
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
ι : Type u_2
𝓕 : Filter ι
μ₀ : ProbabilityMeasure E
μ✝ : ι → ProbabilityMeasure E
h : ∀ (t : E), Tendsto (fun n ↦ charFun (↑(μ✝ n)) t) 𝓕 (𝓝 (charFun (↑μ₀) t))
g : E →ᵇ ℂ
w : AddMonoidAlgebra ℂ E
hw : ⇑g = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(probChar (((innerₗ E) x) a))
μ : Measure E
hμ : IsProbabilityMeasure μ
i : E
hi : i ∈ w.coeff.support
⊢ Integrable (fun x ↦ ↑(probChar (((innerₗ E) x) i))) μ
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case h₁
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))
⊢ μ (⋃ n ∈ Finset.range (N + 1), f n) ≤ μ (⋃ n, f n)

Tactic:
exact measure_mono <| iUnion_subset <|
                      fun i ↦ iUnion_subset fun _ ↦ subset_iUnion f i

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
⊢ μ[|univ] = μ

pp: case hc
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
⊢ IsFiniteMeasure μ

Tactic:
ProbabilityTheory.cond_univ

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
⊢ μ = μ

case hc
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
⊢ IsFiniteMeasure μ
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case ac
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : IsSetAlgebra 𝒜
S : μ.FiniteSpanningSetsIn 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
T : ℕ → Set X := accumulate S.set
T_mem : ∀ (n : ℕ), T n ∈ 𝒜
T_finite : ∀ (n : ℕ), μ (T n) < ∞
T_spanning : ⋃ n, T n = univ
mono : Monotone fun n ↦ T n ∩ s
this✝ : Filter.Tendsto (fun n ↦ ((⇑μ ∘ fun n ↦ T n ∩ s) n).toReal) Filter.atTop (nhds (μ (⋃ n, T n ∩ s)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist ((⇑μ ∘ fun n ↦ T n ∩ s) n).toReal (μ (⋃ n, T n ∩ s)).toReal < ε / 2
this : Fact (μ (T N) < ∞)
t : Set X
t_mem : t ∈ 𝒜
ht : (μ.restrict (T N)) (s ∆ t) < ENNReal.ofReal (ε / 2)
⊢ dist (μ (univ ∩ s)).toReal (μ (T N ∩ s)).toReal < ε / 2

Tactic:
dist_comm,

Goal state after:
case ac
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : IsSetAlgebra 𝒜
S : μ.FiniteSpanningSetsIn 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
T : ℕ → Set X := accumulate S.set
T_mem : ∀ (n : ℕ), T n ∈ 𝒜
T_finite : ∀ (n : ℕ), μ (T n) < ∞
T_spanning : ⋃ n, T n = univ
mono : Monotone fun n ↦ T n ∩ s
this✝ : Filter.Tendsto (fun n ↦ ((⇑μ ∘ fun n ↦ T n ∩ s) n).toReal) Filter.atTop (nhds (μ (⋃ n, T n ∩ s)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist ((⇑μ ∘ fun n ↦ T n ∩ s) n).toReal (μ (⋃ n, T n ∩ s)).toReal < ε / 2
this : Fact (μ (T N) < ∞)
t : Set X
t_mem : t ∈ 𝒜
ht : (μ.restrict (T N)) (s ∆ t) < ENNReal.ofReal (ε / 2)
⊢ dist (μ (T N ∩ s)).toReal (μ (univ ∩ s)).toReal < ε / 2
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case refine_2
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), MeasurableSet s → μ s ≠ ∞ → ∀ (ε : ℝ), 0 < ε → ∃ t ∈ ℬ, μ (s ∆ t) < ENNReal.ofReal ε

Tactic:
intro s ms hμs ε ε_pos

Goal state after:
case refine_2
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
⊢ ∃ t ∈ ℬ, μ (s ∆ t) < ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: 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)
hμ : IsSeparable μ
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
⊢ IsSeparable μ.completion

Tactic:
exact ⟨𝒜, count_𝒜, h𝒜.completion⟩

Goal state after:
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p q : ℝ≥0∞
inst✝² : Fact (1 ≤ q)
ι : Type u_2
inst✝¹ : Fintype ι
E : ι → Type u_3
inst✝ : (i : ι) → NormedAddCommGroup (E i)
f : X → PiLp q E
⊢ MemLp f p μ ↔ MemLp (fun x ↦ (f x).ofLp) p μ

Tactic:
← Function.comp_apply (f := WithLp.ofLp)

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p q : ℝ≥0∞
inst✝² : Fact (1 ≤ q)
ι : Type u_2
inst✝¹ : Fintype ι
E : ι → Type u_3
inst✝ : (i : ι) → NormedAddCommGroup (E i)
f : X → PiLp q E
⊢ MemLp f p μ ↔ MemLp (fun x ↦ (WithLp.ofLp ∘ f) x) p μ
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:

Tactic:
‖a - b‖ * μ.real s ^ (1 / p.toReal)
          ≤ (ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))) * μ.real s ^ (1 / p.toReal) :=
              mul_le_mul_of_nonneg_right (le_of_lt hb) μs_pow_nonneg
        _ ≤ ε / 3 := by
            rw [← mul_one (ε / 3), div_mul_eq_div_mul_one_div, mul_assoc, one_div_mul_eq_div]
            exact mul_le_mul_of_nonneg_left
              ((div_le_one (by linarith [μs_pow_nonneg])).2 (by linarith))
              (by linarith [ε_pos])

Goal state after:
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) := 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 ∈ Q.parts, p ∉ filter MeasurableSet Q.parts → ‖μ p‖ₑ = 0

Tactic:
by simp_all

Goal state after:
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
E : Set X
hE : MeasurableSet E
hE' : ¬⟨E, hE⟩ = ⊥
⊢ ∑ p ∈ (Finpartition.indiscrete hE').parts, ‖μ ↑p‖ₑ ≤ preVariationFun (fun x ↦ ‖μ x‖ₑ) E

Tactic:
by apply preVariation.sum_le

Goal state after:
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 : μ.ennrealVariation s = 0
hsm : MeasurableSet s
this : ‖μ s‖ₑ ≤ 0
⊢ μ s = 0

Tactic:
by simp_all

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

Goal state before:
pp: 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
⊢ μ.IntegrableOn f ∅

Tactic:
by
  simp [VectorMeasure.IntegrableOn]

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

Goal state before:
pp: 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
⊢ μ.IntegrableOn f univ ↔ μ.Integrable f

Tactic:
by
  simp [VectorMeasure.IntegrableOn]

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
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
hs : MeasurableSet s
hfi : μ.IntegrableOn f s
⊢ ∫ᵛ (x : X) in s, s.indicator f x ∂[B; μ] + ∫ᵛ (x : X) in sᶜ, s.indicator f x ∂[B; μ] =
    ∫ᵛ (x : X) in s, f x ∂[B; μ] + ∫ᵛ (x : X) in sᶜ, 0 ∂[B; μ]

Tactic:
apply congr_arg₂ (· + ·) (integral_congr_ae ?_) (integral_congr_ae ?_)
      · rw [variation_restrict hs]
        exact indicator_ae_eq_restrict hs
      · rw [variation_restrict hs.compl]
        exact indicator_ae_eq_restrict_compl hs

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
s 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
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∫ᵛ (x : X) in s, t.indicator f x ∂[B; μ] = ∫ᵛ (x : X) in s ∩ t, f x ∂[B; μ]

Tactic:
rw [integral_indicator ht, μ.restrict_restrict ht hs, Set.inter_comm]

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
s 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
hs : MeasurableSet s
ht : MeasurableSet t
hst : s =ᵐ[μ.variation] t
⊢ s.indicator f =ᵐ[μ.variation] t.indicator f

Tactic:
filter_upwards [hst] with x hx

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
s 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
hs : MeasurableSet s
ht : MeasurableSet t
hst : s =ᵐ[μ.variation] t
x : X
hx : s x = t x
⊢ s.indicator f x = t.indicator f x
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
A : Tendsto (fun n ↦ hf.vectorMeasure (Icc a (u n))) atTop (𝓝 (hf.vectorMeasure (Ici a)))
hu : ∀ (b : α), ∃ a, ∀ (b_1 : ℕ), a ≤ b_1 → b ≤ u b_1
⊢ ∀ᶠ (n : ℕ) in atTop, a ≤ u n

Tactic:
simp [hu]

Goal state after:
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
A : Tendsto (fun n ↦ hf.vectorMeasure (Icc (u n) a)) atTop (𝓝 (hf.vectorMeasure (Iic a)))
hu : ∀ (b : α), ∃ a, ∀ (b_1 : ℕ), a ≤ b_1 → u b_1 ≤ b
⊢ ∀ᶠ (n : ℕ) in atTop, u n ≤ a

Tactic:
simp [hu]

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

Goal state before:
pp: α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : CompleteSpace E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ s ∈ C, MeasurableSet s
M : MeasurableSpace α := generateFrom C
Mle : M ≤ hα
μ' : Measure α := μ.trim Mle
hμ' : μ' = μ.trim Mle
m' : VectorMeasure α E
m'C : ∀ s ∈ C, m' s = m s
hm' : ∀ (s : Set α), ‖m' s‖ₑ ≤ μ' s
m'_le : m'.variation ≤ μ'
this✝ : MeasurableSpace α := hα
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : Pairwise (Function.onFun Disjoint f)
this :
  ∫ᵛ (x : α), μ[fun x ↦ ∑' (d : ℕ), (f d).indicator 1 x | M] x ∂•m' =
    ∫ᵛ (x : α), ∑' (d : ℕ), μ[(f d).indicator 1 | M] x ∂•m'
A : ∀ (i : ℕ), ∫⁻ (a : α), ‖μ[(f i).indicator 1 | M] a‖ₑ ∂μ = μ (f i)
⊢ ∑' (i : ℕ), ∫⁻ (a : α), ‖μ[(f i).indicator 1 | M] a‖ₑ ∂μ < ∞

Tactic:
simp_rw [A, ← measure_iUnion hf f_meas]

Goal state after:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : CompleteSpace E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ s ∈ C, MeasurableSet s
M : MeasurableSpace α := generateFrom C
Mle : M ≤ hα
μ' : Measure α := μ.trim Mle
hμ' : μ' = μ.trim Mle
m' : VectorMeasure α E
m'C : ∀ s ∈ C, m' s = m s
hm' : ∀ (s : Set α), ‖m' s‖ₑ ≤ μ' s
m'_le : m'.variation ≤ μ'
this✝ : MeasurableSpace α := hα
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : Pairwise (Function.onFun Disjoint f)
this :
  ∫ᵛ (x : α), μ[fun x ↦ ∑' (d : ℕ), (f d).indicator 1 x | M] x ∂•m' =
    ∫ᵛ (x : α), ∑' (d : ℕ), μ[(f d).indicator 1 | M] x ∂•m'
A : ∀ (i : ℕ), ∫⁻ (a : α), ‖μ[(f i).indicator 1 | M] a‖ₑ ∂μ = μ (f i)
⊢ μ (⋃ i, f i) < ∞
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: M : Type u_1
ι : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : Finite ι
s : ι → Set M
hs : ∀ (i : ι), IsSemilinearSet (s i)
⊢ IsSemilinearSet (⋃ i, s i)

Tactic:
rw [← sUnion_range]
  apply sUnion (finite_range s)
  simpa

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

Goal state before:
pp: X : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
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
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝ : CompleteSpace G
hf : μ.Integrable f
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = ‖B.flip y‖₊ * ‖x‖₊
s : Set X
hs : MeasurableSet s
ε : NNReal
εpos : 0 < ε
δ : NNReal := ε / 3
δpos : 0 < δ
g : X →ₛ E
hg : eLpNorm (f - ⇑g) 1 (μ.transpose B).variation ≤ ↑δ
gmem : MemLp (⇑g) 1 μ.variation
⊢ ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ ∫⁻ (a : X) in s, ‖f a - g a‖ₑ + ‖g a‖ₑ ∂(μ.transpose B).variation

Tactic:
gcongr with a

Goal state after:
case hfg
X : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
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
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝ : CompleteSpace G
hf : μ.Integrable f
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = ‖B.flip y‖₊ * ‖x‖₊
s : Set X
hs : MeasurableSet s
ε : NNReal
εpos : 0 < ε
δ : NNReal := ε / 3
δpos : 0 < δ
g : X →ₛ E
hg : eLpNorm (f - ⇑g) 1 (μ.transpose B).variation ≤ ↑δ
gmem : MemLp (⇑g) 1 μ.variation
a : X
⊢ ‖f a‖ₑ ≤ ‖f a - g a‖ₑ + ‖g a‖ₑ
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Definability

Goal state before:
pp: α : Type u_1
A : Set ℕ
inst✝ : Finite α
n✝ : ℕ
this : Fintype α
n : ℕ
φ : presburger[[↑A]].BoundedFormula α (n + 1)
e : (α ⊕ Fin n) ⊕ Fin 1 ≃ α ⊕ Fin (n + 1) :=
  (Equiv.sumAssoc α (Fin n) (Fin 1)).trans ((_root_.Equiv.refl α).sumCongr finSumFinEquiv)
ih : IsSemilinearSet (⇑(LinearEquiv.funCongrLeft ℕ ℕ e) '' {v | φ.Realize (v ∘ Sum.inl) (v ∘ Sum.inr)})
⊢ {v | ∀ (x : ℕ), φ.Realize (v ∘ Sum.inl) (Fin.snoc (v ∘ Sum.inr) x)} =
    {x | ∃ y, Sum.elim x y ∈ (⇑(LinearEquiv.funCongrLeft ℕ ℕ e) '' {v | φ.Realize (v ∘ Sum.inl) (v ∘ Sum.inr)})ᶜ}ᶜ

Tactic:
simp_rw [compl_ofPred, not_exists, Fin.forall_fin_succ_pi, Fin.forall_fin_zero_pi,
      mem_compl_iff, mem_image, not_not, ← LinearEquiv.eq_symm_apply, LinearEquiv.funCongrLeft_symm,
      exists_eq_right, mem_ofPred, LinearEquiv.funCongrLeft_apply, LinearMap.funLeft,
      LinearMap.coe_mk, AddHom.coe_mk]

Goal state after:
α : Type u_1
A : Set ℕ
inst✝ : Finite α
n✝ : ℕ
this : Fintype α
n : ℕ
φ : presburger[[↑A]].BoundedFormula α (n + 1)
e : (α ⊕ Fin n) ⊕ Fin 1 ≃ α ⊕ Fin (n + 1) :=
  (Equiv.sumAssoc α (Fin n) (Fin 1)).trans ((_root_.Equiv.refl α).sumCongr finSumFinEquiv)
ih : IsSemilinearSet (⇑(LinearEquiv.funCongrLeft ℕ ℕ e) '' {v | φ.Realize (v ∘ Sum.inl) (v ∘ Sum.inr)})
⊢ {v | ∀ (x : ℕ), φ.Realize (v ∘ Sum.inl) (Fin.snoc (v ∘ Sum.inr) x)} =
    {a |
      ∀ (a_1 : ℕ),
        φ.Realize ((Sum.elim a (Fin.cons a_1 finZeroElim) ∘ ⇑e.symm) ∘ Sum.inl)
          ((Sum.elim a (Fin.cons a_1 finZeroElim) ∘ ⇑e.symm) ∘ Sum.inr)}