mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar

Goal state before:
pp: G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
φ : G ≃ₜ* G
e : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv
⊢ mulEquivHaarChar φ • μ = comap (⇑e) μ

Tactic:
← e.map_symm,

Goal state after:
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
φ : G ≃ₜ* G
e : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv
⊢ mulEquivHaarChar φ • μ = map (⇑e.symm) μ
Module: Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar

Goal state before:
pp: G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
f : G → ℝ
φ : G ≃ₜ* G
e : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv
this✝ : (map (⇑e.symm) μ).IsHaarMeasure
this : (map (⇑e.symm) μ).Regular
⊢ ∫ (a : G), f a ∂comap (⇑e) μ = mulEquivHaarChar φ • ∫ (a : G), f a ∂μ

Tactic:
← e.map_symm,

Goal state after:
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
f : G → ℝ
φ : G ≃ₜ* G
e : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv
this✝ : (map (⇑e.symm) μ).IsHaarMeasure
this : (map (⇑e.symm) μ).Regular
⊢ ∫ (a : G), f a ∂map (⇑e.symm) μ = mulEquivHaarChar φ • ∫ (a : G), f a ∂μ
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
x : ι → ℝ
⊢ ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)

Tactic:
simp [PiLp.norm_eq_sum (f := toLp (.ofReal p) x) (this.symm ▸ h₁), this]

Goal state after:
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
x : ι → ℝ
⊢ ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)

Tactic:
simp [PiLp.norm_eq_sum (f := toLp (.ofReal p) x) (this.symm ▸ h₁), this]

Goal state after:
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 → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
x : ι → ℂ
⊢ ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)

Tactic:
simp [PiLp.norm_eq_sum (f := toLp (.ofReal p) x) (this.symm ▸ h₁), this]

Goal state after:
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 → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
⊢ volume {x | ∑ i, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Tactic:
have : Fact (1 ≤ ENNReal.ofReal p) := fact_iff.mpr (ENNReal.ofReal_one ▸ (ofReal_le_ofReal hp))

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 → 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)
⊢ volume {x | ∑ i, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(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
x : ι → ℂ
⊢ ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)

Tactic:
simp [PiLp.norm_eq_sum (f := toLp (.ofReal p) x) (this.symm ▸ h₁), this]

Goal state after:
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 → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
⊢ volume {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) ≤ r} =
    ENNReal.ofReal r ^ (2 * card ι) *
      ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Tactic:
have : Fact (1 ≤ ENNReal.ofReal p) := fact_iff.mpr (ENNReal.ofReal_one ▸ (ofReal_le_ofReal hp))

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 → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
⊢ volume {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) ≤ r} =
    ENNReal.ofReal r ^ (2 * card ι) *
      ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: x : EuclideanSpace ℝ (Fin 3)
r : ℝ
⊢ volume (closedBall x r) = ENNReal.ofReal r ^ 3 * ENNReal.ofReal (π * 4 / 3)

Tactic:
rw [addHaar_closedBall_eq_addHaar_ball, volume_ball_fin_three x]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s t : MeasuredSets μ
⊢ μ ↑s ≤ μ ↑t + edist s t

Tactic:
← tsub_le_iff_left

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s t : MeasuredSets μ
⊢ μ ↑s - μ ↑t ≤ edist s t
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s₁ s₂ : Set X
hs₂ : MeasurableSet s₂
h : s₁ ⊆ s₂
⊢ preVariationFun f s₁ ≤ preVariationFun f s₂

Tactic:
by_cases hs₁ : MeasurableSet s₁

Goal state after:
case pos
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s₁ s₂ : Set X
hs₂ : MeasurableSet s₂
h : s₁ ⊆ s₂
hs₁ : MeasurableSet s₁
⊢ preVariationFun f s₁ ≤ preVariationFun f s₂

case neg
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s₁ s₂ : Set X
hs₂ : MeasurableSet s₂
h : s₁ ⊆ s₂
hs₁ : ¬MeasurableSet s₁
⊢ preVariationFun f s₁ ≤ preVariationFun f s₂
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: case pos
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s₁ s₂ : Set X
hs₂ : MeasurableSet s₂
h : s₁ ⊆ s₂
hs₁ : MeasurableSet s₁
⊢ preVariationFun f s₁ ≤ preVariationFun f s₂

Tactic:
have := sum_le_preVariationFun_of_subset f hs₁ hs₂ h

Goal state after:
case pos
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s₁ s₂ : Set X
hs₂ : MeasurableSet s₂
h : s₁ ⊆ s₂
hs₁ : MeasurableSet s₁
this : ∀ (P : Finpartition ⟨s₁, hs₁⟩), ∑ p ∈ P.parts, f ↑p ≤ preVariationFun f s₂
⊢ preVariationFun f s₁ ≤ preVariationFun f s₂
Module: Mathlib.MeasureTheory.Measure.ResolventTransform

Goal state before:
pp: 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
⊢ infDist a (⇑(algebraMap 𝕜 A) '' μ.support) - infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2 ≤
    ‖(algebraMap 𝕜 A) x - a‖ - ‖a - w‖

Tactic:
gcongr
        · grw [infDist_le_dist_of_mem (y := (algebraMap 𝕜 A) x) (by simp [hx]), dist_comm,
            dist_eq_norm]
        · apply le_of_lt
          simpa [s, ← dist_eq_norm, dist_comm w a] using hw

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
this✝ : μ (s ∩ t) ≤ μ s
this : μ (s ∩ t) ≤ μ t
⊢ μ (s ∩ t) = μ s * μ t

Tactic:
cases μ.zero_one s

Goal state after:
case inl
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
this✝ : μ (s ∩ t) ≤ μ s
this : μ (s ∩ t) ≤ μ t
h✝ : μ s = 0
⊢ μ (s ∩ t) = μ s * μ t

case inr
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
this✝ : μ (s ∩ t) ≤ μ s
this : μ (s ∩ t) ≤ μ t
h✝ : μ s = 1
⊢ μ (s ∩ t) = μ s * μ t
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝³ : TopologicalSpace V
inst✝² : ENormedAddCommMonoid V
inst✝¹ : T2Space V
μ : VectorMeasure X V
Y : Type u_3
inst✝ : MeasurableSpace Y
φ : X → Y
hφ : MeasurableEmbedding φ
s : Set Y
hs : MeasurableSet s
this : (μ.map φ).variation (s \ Set.range φ) = 0
⊢ (μ.map φ).variation (s ∩ Set.range φ ∪ s \ Set.range φ) = (μ.map φ).variation (s ∩ Set.range φ)

Tactic:
measure_union (by grind) (hs.diff hφ.measurableSet_range),

Goal state after:
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝³ : TopologicalSpace V
inst✝² : ENormedAddCommMonoid V
inst✝¹ : T2Space V
μ : VectorMeasure X V
Y : Type u_3
inst✝ : MeasurableSpace Y
φ : X → Y
hφ : MeasurableEmbedding φ
s : Set Y
hs : MeasurableSet s
this : (μ.map φ).variation (s \ Set.range φ) = 0
⊢ (μ.map φ).variation (s ∩ Set.range φ) + (μ.map φ).variation (s \ Set.range φ) = (μ.map φ).variation (s ∩ Set.range φ)
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 : ∀ i ∈ insert a✝¹ s✝, MeasurableSet (t i)
h't : ∀ i ∈ insert a✝¹ s✝, μ.IntegrableOn f (t i)
⊢ μ.IntegrableOn f (⋃ i ∈ insert a✝¹ s✝, t i)

Tactic:
simp only [mem_insert_iff, forall_eq_or_imp, iUnion_iUnion_eq_or_left] at ht h't ⊢

Goal state after:
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)
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)
A : ∀ (n : ℕ), m (⋃ i ∈ Finset.range n, f i) = ∑ i ∈ Finset.range n, m (f i)
B : ∀ (n : ℕ), m (⋃ i, f i) = m (⋃ i ∈ Finset.range n, f i) + m (⋃ i ∈ Set.Ici n, f i)
C : ∀ (n : ℕ), m (⋃ i, f i) - ∑ i ∈ Finset.range n, m (f i) = m (⋃ i ∈ Set.Ici n, f i)
⊢ Filter.Tendsto (fun n ↦ μ (⋃ i ∈ Set.Ici n, f i)) Filter.atTop (nhds ⊥)

Tactic:
exact tendsto_measure_biUnion_Ici_zero_of_pairwise_disjoint
      (fun i ↦ (f_meas i).nullMeasurableSet) f_disj

Goal state after:
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
s : Set X
hs : MeasurableSet s
h's : μ.semivariation s = ∞
t : Set X
ts : t ⊆ s
t_meas : MeasurableSet t
ht : 2 * ‖μ s‖ₑ + 2 < 2 * ‖μ t‖ₑ
⊢ 1 + ‖μ s‖ₑ ≤ ‖μ t‖ₑ

Tactic:
apply (ENNReal.mul_le_mul_iff_right (a := 2) (by simp) (by simp)).1

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
s : Set X
hs : MeasurableSet s
h's : μ.semivariation s = ∞
t : Set X
ts : t ⊆ s
t_meas : MeasurableSet t
ht : 2 * ‖μ s‖ₑ + 2 < 2 * ‖μ t‖ₑ
⊢ 2 * (1 + ‖μ s‖ₑ) ≤ 2 * ‖μ t‖ₑ
Module: Mathlib.MeasureTheory.VectorMeasure.AddContent

Goal state before:
pp: case hm
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 : IsSetRing 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 ∈ C, μ (s ∆ t) < ε
C' : Set (MeasuredSets μ) := ⋯
C'C : ∀ s ∈ C', ↑s ∈ C
C'_dense : Dense C'
m₀ : ↑C' → E := ⋯
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := ⋯
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ μ ↑s
hAddit : ∀ (s t : MeasuredSets μ), Disjoint ↑s ↑t → m₁ ⟨↑s ∪ ↑t, ⋯⟩ = m₁ s + m₁ t
m' : Set α → E := ⋯
⊢ ∀ (s : Set α), ‖m' s‖ₑ ≤ μ s

Tactic:
intro s
      by_cases hs : MeasurableSet s
      · simpa [hs, m'] using! hBound _
      · simp [hs, m']

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

Goal state before:
pp: case h'm
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 : IsSetRing 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 ∈ C, μ (s ∆ t) < ε
C' : Set (MeasuredSets μ) := ⋯
C'C : ∀ s ∈ C', ↑s ∈ C
C'_dense : Dense C'
m₀ : ↑C' → E := ⋯
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := ⋯
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ μ ↑s
hAddit : ∀ (s t : MeasuredSets μ), Disjoint ↑s ↑t → m₁ ⟨↑s ∪ ↑t, ⋯⟩ = m₁ s + m₁ t
m' : Set α → E := ⋯
⊢ ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m' (s ∪ t) = m' s + m' t

pp: case h''m
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 : IsSetRing 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 ∈ C, μ (s ∆ t) < ε
C' : Set (MeasuredSets μ) := ⋯
C'C : ∀ s ∈ C', ↑s ∈ C
C'_dense : Dense C'
m₀ : ↑C' → E := ⋯
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := ⋯
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ μ ↑s
hAddit : ∀ (s t : MeasuredSets μ), Disjoint ↑s ↑t → m₁ ⟨↑s ∪ ↑t, ⋯⟩ = m₁ s + m₁ t
m' : Set α → E := ⋯
⊢ ∀ (s : Set α), ¬MeasurableSet s → m' s = 0

Tactic:
· intro s t hs ht hst
      simp only [hs, ht, MeasurableSet.union, ↓reduceDIte, m']
      exact hAddit ⟨s, hs⟩ ⟨t, ht⟩ hst

Goal state after:
case h''m
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 : IsSetRing 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 ∈ C, μ (s ∆ t) < ε
C' : Set (MeasuredSets μ) := ⋯
C'C : ∀ s ∈ C', ↑s ∈ C
C'_dense : Dense C'
m₀ : ↑C' → E := ⋯
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := ⋯
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ μ ↑s
hAddit : ∀ (s t : MeasuredSets μ), Disjoint ↑s ↑t → m₁ ⟨↑s ∪ ↑t, ⋯⟩ = m₁ s + m₁ t
m' : Set α → E := ⋯
⊢ ∀ (s : Set α), ¬MeasurableSet s → m' s = 0
Module: Mathlib.MeasureTheory.VectorMeasure.AddContent

Goal state before:
pp: case refine_1
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 : IsSetRing 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 ∈ C, μ (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s | ∃ c ∈ C, ↑s = c}
C'C : ∀ s ∈ C', ↑s ∈ C
C'_dense : Dense C'
m₀ : ↑C' → E := fun x ↦ m ↑↑x
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ μ ↑s
hAddit : ∀ (s t : MeasuredSets μ), Disjoint ↑s ↑t → m₁ ⟨↑s ∪ ↑t, ⋯⟩ = m₁ s + m₁ t
m' : Set α → E := fun s ↦ if hs : MeasurableSet s then m₁ ⟨s, hs⟩ else 0
m'' : VectorMeasure α E := of_additive_of_le_measure m' ⋯ ⋯ ⋯
s : Set α
hs : s ∈ C
⊢ m' s = m s

Tactic:
simp only [hCmeas s hs, ↓reduceDIte, m']

Goal state after:
case refine_1
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 : IsSetRing 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 ∈ C, μ (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s | ∃ c ∈ C, ↑s = c}
C'C : ∀ s ∈ C', ↑s ∈ C
C'_dense : Dense C'
m₀ : ↑C' → E := fun x ↦ m ↑↑x
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ μ ↑s
hAddit : ∀ (s t : MeasuredSets μ), Disjoint ↑s ↑t → m₁ ⟨↑s ∪ ↑t, ⋯⟩ = m₁ s + m₁ t
m' : Set α → E := fun s ↦ if hs : MeasurableSet s then m₁ ⟨s, hs⟩ else 0
m'' : VectorMeasure α E := of_additive_of_le_measure m' ⋯ ⋯ ⋯
s : Set α
hs : s ∈ C
⊢ m₁ ⟨s, ⋯⟩ = m s
Module: Mathlib.MeasureTheory.VectorMeasure.Prod

Goal state before:
pp: case compl
X : Type u_2
Y : Type u_3
F : Type u_5
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝ : NormedAddCommGroup F
ν : VectorMeasure Y F
s✝ s : Set (X × Y)
hs : MeasurableSet s
ihs : StronglyMeasurable fun x ↦ ν (Prod.mk x ⁻¹' s)
⊢ StronglyMeasurable fun x ↦ ν (Prod.mk x ⁻¹' sᶜ)

Tactic:
simp_rw [preimage_compl, VectorMeasure.of_compl (measurable_prodMk_left hs)]

Goal state after:
case compl
X : Type u_2
Y : Type u_3
F : Type u_5
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝ : NormedAddCommGroup F
ν : VectorMeasure Y F
s✝ s : Set (X × Y)
hs : MeasurableSet s
ihs : StronglyMeasurable fun x ↦ ν (Prod.mk x ⁻¹' s)
⊢ StronglyMeasurable fun x ↦ ν univ - ν (Prod.mk x ⁻¹' s)
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos.h_closed.hf
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
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun y ↦ ∫ᵛ (y_1 : X), C (↑↑y y_1) ∂[B; μ]

Tactic:
have I (f : Lp H 1 μ.variation) : ∫ᵛ x, C (f x) ∂[B; μ] = ∫ᵛ x, (C.compLp f) x ∂[B; μ] :=
        (integral_congr_ae (coeFn_compLp _ _)).symm

Goal state after:
case pos.h_closed.hf
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
I : ∀ (f : ↥(Lp H 1 μ.variation)), ∫ᵛ (x : X), C (↑↑f x) ∂[B; μ] = ∫ᵛ (x : X), ↑↑(C.compLp f) x ∂[B; μ]
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun y ↦ ∫ᵛ (y_1 : X), C (↑↑y y_1) ∂[B; μ]
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
C : ℝ
h : IsFiniteMeasure (μ.variation.restrict s)
hC : ∀ᵐ (x : X) ∂μ.variation.restrict s, ‖f x‖ ≤ C
hs : ¬MeasurableSet s
h's : ¬μ.variation s = 0
⊢ (ae (μ.variation.restrict s)).NeBot

Tactic:
simpa using h's

Goal state after:
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Basic

Goal state before:
pp: α : Type u_1
M : Type u_2
v : α → M
inst✝ : AddCommMonoidWithOne M
β : Type u_3
s : Finset β
f : β → presburger.Term α
⊢ Term.realize v (sum s f) = ∑ i ∈ s, Term.realize v (f i)

Tactic:
by
  classical
  simp only [sum]
  conv => rhs; rw [← s.toList_toFinset, List.sum_toFinset _ s.nodup_toList]
  generalize s.toList = l
  induction l with simp [*]

Goal state after:
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✝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
B : E →L[ℝ] F →L[ℝ] G
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ ∫ᵛ (a : X), ↑↑f a ∂[B; μ]

Tactic:
by
  simp only [integral_eq_setToFun]
  exact continuous_setToFun _

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

Goal state before:
pp: case inr
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
ι : Type u_7
inst✝ : Countable ι
s : ι → Set X
hm : ∀ (i : ι), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hι : Infinite ι
e : ι ≃ ℕ
⊢ HasSum (fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]) ∫ᵛ (x : X) in ⋃ n, s n, f x ∂[B; μ]

Tactic:
rw [← e.symm.surjective.iUnion_comp, ← e.symm.hasSum_iff]

Goal state after:
case inr
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
ι : Type u_7
inst✝ : Countable ι
s : ι → Set X
hm : ∀ (i : ι), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hι : Infinite ι
e : ι ≃ ℕ
⊢ HasSum ((fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]) ∘ ⇑e.symm) ∫ᵛ (x : X) in ⋃ x, s (e.symm x), f x ∂[B; μ]
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: ι : Type u_1
inst✝ : Finite ι
s : Set (ι → ℕ)
hs : IsSlice s
h : ∀ (a : ι → ℕ) (t : Finset ι), (∀ x ∈ s, ∀ i ∉ t, x i = a i) → IsSemilinearSet s
this : Fintype ι
⊢ IsSemilinearSet s

Tactic:
exact h 0 Finset.univ (by simp)

Goal state after:
Module: Mathlib.ModelTheory.Complexity

Goal state before:
pp: case of_isQF
L : Language
α : Type u'
n : ℕ
M : Type u_1
inst✝⁴ : L.Structure M
N : Type u_2
inst✝³ : L.Structure N
F : Type u_3
inst✝² : FunLike F M N
inst✝¹ : EmbeddingLike F M N
inst✝ : L.StrongHomClass F M N
φ : L.BoundedFormula α n
f : F
v : α → M
n✝ : ℕ
φ✝ : L.BoundedFormula α n✝
hQF : φ✝.IsQF
xs : Fin n✝ → M
⊢ φ✝.Realize (⇑f ∘ v) (⇑f ∘ xs) → φ✝.Realize v xs

Tactic:
| of_isQF hQF =>

Goal state after: