mathlib-initiative/mathlib-tactics huggingface dataset:
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
μ : FiniteMeasure E
n : ℕ
⊢ ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)

Tactic:
← biUnion_range_succ_disjointed,

Goal state after:
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
μ : FiniteMeasure E
n : ℕ
⊢ ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict (⋃ i ∈ Finset.range (n + 1), disjointed K i)
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
n : ℕ
ρ : FiniteMeasure E
hρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), ρ (K n)ᶜ ≤ u n
⊢ Disjoint ((partialSups K) n)ᶜ (disjointed K n)

Tactic:
apply disjoint_compl_left.mono_right

Goal state after:
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
n : ℕ
ρ : FiniteMeasure E
hρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), ρ (K n)ᶜ ≤ u n
⊢ disjointed K n ⊆ (partialSups K) n
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 = ↑(∑ i, 2) / p + 1

Tactic:
Finset.sum_const,

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 * ↑(card ι) / p + 1 = ↑(Finset.univ.card • 2) / p + 1
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case e'_3
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
h₂ : ∀ (x : ι → ℂ), 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p)
hr : 0 < r
⊢ ENNReal.ofReal (r ^ (2 * card ι)) = ENNReal.ofReal (r ^ ∑ x, 2)

Tactic:
Finset.sum_const,

Goal state after:
case e'_3
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
h₂ : ∀ (x : ι → ℂ), 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p)
hr : 0 < r
⊢ ENNReal.ofReal (r ^ (2 * card ι)) = ENNReal.ofReal (r ^ (Finset.univ.card • 2))
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case e'_3
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
hφs : range ⇑φ ⊆ s
this :
  ∫ᶜ (x : E) in Path.segment a b, ω x + ∫ᶜ (x : E) in φ.evalAt 1, ω x =
    ∫ᶜ (x : E) in Path.segment a c, ω x + ∫ᶜ (x : E) in φ.evalAt 0, ω x
⊢ ∫ᶜ (x : E) in Path.segment a c, ω x = ∫ᶜ (x : E) in Path.segment a c, ω x + ∫ᶜ (x : E) in φ.evalAt 0, ω x

Tactic:
· dsimp [φ]
      rw [← Path.cast_segment (lineMap_apply_zero a b) (lineMap_apply_zero a c)]
      simp

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
a b : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
inst✝ : CompleteSpace F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
⊢ HasFDerivWithinAt (fun x ↦ ∫ᶜ (x : E) in Path.segment a x, ω x) (ω b) s b

Tactic:
suffices HasFDerivWithinAt (∫ᶜ x in .segment a b, ω x + ∫ᶜ x in .segment b ·, ω x) (ω b) s b from
    this.congr' (fun _ h ↦
      (hs.curveIntegral_segment_add_eq_of_hasFDerivWithinAt_symmetric hω hdω ha hb h).symm) hb
  refine .const_add _ <| ?_
  refine HasFDerivWithinAt.curveIntegral_segment_source hs ?_ hb
  exact fun x hx ↦ (hω x hx).continuousWithinAt

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

Goal state before:
pp: case hbc
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
h_le_Ioc : ∀ x ∈ Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ‖1 - f (x • z)‖ ≤ ε / 4
⊢ ∫ (x : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ‖1 - f (x • z)‖ ∂volume ≤
    ∫ (x : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ε / 4 ∂volume

Tactic:
have hf_meas : Measurable f := by
      refine measurable_of_tendsto_metrizable (f := fun n t ↦ charFun (μ n) t) (by fun_prop) ?_
      rwa [tendsto_pi_nhds]

Goal state after:
case hbc
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
h_le_Ioc : ∀ x ∈ Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ‖1 - f (x • z)‖ ≤ ε / 4
hf_meas : Measurable f
⊢ ∫ (x : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ‖1 - f (x • z)‖ ∂volume ≤
    ∫ (x : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ε / 4 ∂volume
Module: Mathlib.MeasureTheory.Measure.ResolventTransform

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : CompleteSpace 𝕜
inst✝¹ : NormedDivisionRing A
inst✝ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
x : 𝕜
hx : x ∈ μ.support
⊢ 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)

Tactic:
refine (IsClosed.notMem_iff_infDist_pos ?_ ((Set.nonempty_of_mem hx).image _)).mp hz

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : CompleteSpace 𝕜
inst✝¹ : NormedDivisionRing A
inst✝ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
x : 𝕜
hx : x ∈ μ.support
⊢ IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑(algebraMap 𝕜 A) '' μ.support)
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ i, f i
A : ℕ → Finset (Set α) := fun i ↦ hC.disjointOfDiff ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ ↑(A i)
h_m_eq : ∀ (i : ℕ), m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u
⊢ (OuterMeasure.ofFunction ⇑m ⋯) (t ∩ s) + (OuterMeasure.ofFunction ⇑m ⋯) (t \ s) ≤
    ∑' (a : ℕ), m (f a ∩ s) + ∑' (a : ℕ), ∑ u ∈ A a, m u

Tactic:
refine add_le_add ?_ ?_

Goal state after:
case refine_1
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ i, f i
A : ℕ → Finset (Set α) := fun i ↦ hC.disjointOfDiff ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ ↑(A i)
h_m_eq : ∀ (i : ℕ), m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u
⊢ (OuterMeasure.ofFunction ⇑m ⋯) (t ∩ s) ≤ ∑' (a : ℕ), m (f a ∩ s)

case refine_2
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ i, f i
A : ℕ → Finset (Set α) := fun i ↦ hC.disjointOfDiff ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ ↑(A i)
h_m_eq : ∀ (i : ℕ), m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u
⊢ (OuterMeasure.ofFunction ⇑m ⋯) (t \ s) ≤ ∑' (a : ℕ), ∑ u ∈ A a, m u
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: case pos
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
⊢ ‖μ s‖ₑ ≤ 0

Tactic:
enorm_measure_le_variation,

Goal state after:
case pos
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
⊢ μ.variation s ≤ 0
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: case pos
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
⊢ μ s = 0

pp: case neg
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
⊢ μ s = 0

Tactic:
· suffices ‖μ s‖ₑ ≤ 0 by simp_all
    grw [enorm_measure_le_variation, ← ennrealVariation_apply _ hsm, hs]

Goal state after:
case neg
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
⊢ μ s = 0
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
⊢ μ ≪ᵥ μ.ennrealVariation

Tactic:
intro s hs
  by_cases hsm : MeasurableSet s
  · suffices ‖μ s‖ₑ ≤ 0 by simp_all
    grw [enorm_measure_le_variation, ← ennrealVariation_apply _ hsm, hs]
  · exact μ.not_measurable hsm

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
m : Measure X
hs : MeasurableSet s
h : ∀ (E : Set X), MeasurableSet E → E ⊆ s → ‖μ E‖ₑ ≤ m E
⊢ μ.variation s ≤ m s

Tactic:
simp only [variation_apply, preVariation, ennrealToMeasure_apply hs, ennrealPreVariation_apply,
    preVariationFun, hs, dite_true, iSup_le_iff]

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
m : Measure X
hs : MeasurableSet s
h : ∀ (E : Set X), MeasurableSet E → E ⊆ s → ‖μ E‖ₑ ≤ m E
⊢ ∀ (i : Finpartition ⟨s, ⋯⟩), ∑ x ∈ i.parts, ‖μ ↑x‖ₑ ≤ m s
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup V
μ : VectorMeasure X V
𝕜 : Type u_3
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
c : 𝕜
s : Set X
hs : MeasurableSet s
⊢ ‖c‖ₑ * ‖μ s‖ₑ ≤ ↑‖c‖₊ * μ.variation s

Tactic:
enorm_measure_le_variation,

Goal state after:
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup V
μ : VectorMeasure X V
𝕜 : Type u_3
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
c : 𝕜
s : Set X
hs : MeasurableSet s
⊢ ‖c‖ₑ * μ.variation s ≤ ↑‖c‖₊ * μ.variation 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
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : hα = generateFrom C
h''C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
s : Set α
hs : s ∈ C
⊢ MeasurableSet s

Tactic:
rw [h'C]

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
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : hα = generateFrom C
h''C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
s : Set α
hs : s ∈ C
⊢ MeasurableSet 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
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : hα = generateFrom C
s : Set α
hs : s ∈ C
⊢ MeasurableSet s

Tactic:
rw [h'C]

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
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : hα = generateFrom C
s : Set α
hs : s ∈ C
⊢ MeasurableSet s
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
m'' : VectorMeasure α E :=
  { measureOf' := fun s ↦ if MeasurableSet s then ∫ᵛ (x : α), μ[s.indicator 1 | M] x ∂•m' else 0, empty' := ⋯,
    not_measurable' := ⋯, m_iUnion' := ⋯ }
s : Set α
hs : s ∈ C
x : α
⊢ μ[s.indicator 1 | M] x = s.indicator 1 x

Tactic:
rw [condExp_of_stronglyMeasurable Mle]

Goal state after:
case hf
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
m'' : VectorMeasure α E :=
  { measureOf' := fun s ↦ if MeasurableSet s then ∫ᵛ (x : α), μ[s.indicator 1 | M] x ∂•m' else 0, empty' := ⋯,
    not_measurable' := ⋯, m_iUnion' := ⋯ }
s : Set α
hs : s ∈ C
x : α
⊢ StronglyMeasurable (s.indicator 1)

case hfi
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
m'' : VectorMeasure α E :=
  { measureOf' := fun s ↦ if MeasurableSet s then ∫ᵛ (x : α), μ[s.indicator 1 | M] x ∂•m' else 0, empty' := ⋯,
    not_measurable' := ⋯, m_iUnion' := ⋯ }
s : Set α
hs : s ∈ C
x : α
⊢ Integrable (s.indicator 1) μ
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
s : Set X
hs : MeasurableSet s
x : E
⊢ ‖B‖ₑ * ‖x‖ₑ * ‖μ s‖ₑ ≤ ↑‖B‖₊ * μ.variation s * ‖x‖ₑ

Tactic:
enorm_measure_le_variation,

Goal state after:
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
s : Set X
hs : MeasurableSet s
x : E
⊢ ‖B‖ₑ * ‖x‖ₑ * μ.variation s ≤ ↑‖B‖₊ * μ.variation s * ‖x‖ₑ
Module: Mathlib.MeasureTheory.VectorMeasure.Integral

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : Nontrivial E
C : ℝ≥0
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = C * ‖x‖₊ * ‖y‖₊
hC : C ≠ 0
s : Set X
hs : MeasurableSet s
this : ‖μ s‖ₑ ≤ C⁻¹ • ‖(μ.transpose B) s‖ₑ
⊢ C⁻¹ • ‖(μ.transpose B) s‖ₑ ≤ (C⁻¹ • (μ.transpose B).variation) s

Tactic:
enorm_measure_le_variation,

Goal state after:
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : Nontrivial E
C : ℝ≥0
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = C * ‖x‖₊ * ‖y‖₊
hC : C ≠ 0
s : Set X
hs : MeasurableSet s
this : ‖μ s‖ₑ ≤ C⁻¹ • ‖(μ.transpose B) s‖ₑ
⊢ C⁻¹ • (μ.transpose B).variation s ≤ (C⁻¹ • (μ.transpose B).variation) s
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: case mpr.inr
M : Type u_1
inst✝ : AddCommMonoid M
a : M
t : Set M
ht : t.Finite
x : M
hx : ∃ y ∈ closure {a}, ∃ z ∈ closure t, y + z = x
⊢ a + x ∈ closure (a +ᵥ ↑(closure t))

Tactic:
mem_closure_singleton

Goal state after:
case mpr.inr
M : Type u_1
inst✝ : AddCommMonoid M
a : M
t : Set M
ht : t.Finite
x : M
hx : ∃ y, (∃ n, n • a = y) ∧ ∃ z ∈ closure t, y + z = x
⊢ a + x ∈ closure (a +ᵥ ↑(closure t))
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✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
f : X → E
B : E →L[ℝ] F →L[ℝ] G
hf : μ.Integrable f
s : Set X
hs : MeasurableSet s
⊢ ‖(μ.withDensity f B) s‖ₑ ≤ ((μ.transpose B).variation.withDensity fun x ↦ ‖f x‖ₑ) s

Tactic:
rw [withDensity_apply hf, MeasureTheory.withDensity_apply _ hs]

Goal state after:
X : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
f : X → E
B : E →L[ℝ] F →L[ℝ] G
hf : μ.Integrable f
s : Set X
hs : MeasurableSet s
⊢ ‖∫ᵛ (x : X) in s, f x ∂[B; μ]‖ₑ ≤ ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation