mathlib-initiative/mathlib-tactics huggingface dataset:
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✝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)
⊢ 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:
by

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

Goal state before:
pp: S : Finset (Set ℕ)
hS : ∀ t ∈ S, IsProperLinearSet t
⊢ ∀ t ∈ S, ∃ k, ∃ p > 0, ∀ x ≥ k, x ∈ t ↔ x + p ∈ t

Tactic:
by

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

Goal state before:
pp: X : Type u_2
G : Type u_5
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : Measure X
inst✝ : IsFiniteMeasure μ
f : X → G
s : Set X
hs : MeasurableSet s
⊢ ∫ᵛ (x : X) in s, f x ∂<•μ.toSignedMeasure = ∫ᵛ (x : X), f x ∂<•(μ.restrict s).toSignedMeasure

Tactic:
restrict_toSignedMeasure hs

Goal state after:
X : Type u_2
G : Type u_5
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : Measure X
inst✝ : IsFiniteMeasure μ
f : X → G
s : Set X
hs : MeasurableSet s
⊢ ∫ᵛ (x : X), f x ∂<•(μ.restrict s).toSignedMeasure = ∫ᵛ (x : X), f x ∂<•(μ.restrict s).toSignedMeasure
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: X : Type u_2
G : Type u_5
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : Measure X
inst✝ : IsFiniteMeasure μ
f : X → G
s : Set X
hs : MeasurableSet s
⊢ ∫ᵛ (x : X) in s, f x ∂<•μ.toSignedMeasure = ∫ (x : X) in s, f x ∂μ

Tactic:
by

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
f g : X → E
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
hf : μ.Integrable f
hg : μ.Integrable g
⊢ edist ∫ᵛ (a : X), f a ∂[B; μ] ∫ᵛ (a : X), g a ∂[B; μ] ≤ ‖B‖ₑ * ∫⁻ (a : X), edist (f a) (g a) ∂μ.variation

Tactic:
by

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
I1 : ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation + ↑δ
I2 :
  ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation =
    ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i})
ρ : ℝ≥0∞
ρpos : ρ > 0
hρ : ∑ i ∈ g.range, ‖i‖ₑ * ρ ≤ ↑δ
i : E
hi : i ≠ 0
P : Finset (Set X)
hP : ∀ t ∈ P, t ⊆ ⇑g ⁻¹' {i}
h'P : (↑P).PairwiseDisjoint id
h''P : ∀ t ∈ P, MeasurableSet t
h'''P : ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤ ∑ p ∈ P, ‖((μ.transpose B).restrict s) p‖ₑ + ρ
⊢ ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤ ‖i‖ₑ * (∑ p ∈ P, ‖((μ.transpose B).restrict s) p‖ₑ + ρ)

Tactic:
by

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
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
ι : Type u_7
hf : μ.Integrable f
l : Filter ι
s : ι → Set X
hs : Tendsto (⇑μ.variation ∘ s) l (𝓝 0)
⊢ 0 ≠ 0 ∨ ‖B‖ₑ ≠ ∞

Tactic:
by

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

Goal state before:
pp: X : Type u_2
G : Type u_6
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : Measure X
inst✝ : IsFiniteMeasure μ
f : X → G
⊢ ∫ᵛ (x : X), f x ∂<•μ.toSignedMeasure = ∫ (x : X), f x ∂μ

Tactic:
by

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
I1 : ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation + ↑δ
I2 :
  ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation =
    ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i})
ρ : ℝ≥0∞
ρpos : ρ > 0
hρ : ∑ i ∈ g.range, ‖i‖ₑ * ρ ≤ ↑δ
i : E
hi : i ≠ 0
this :
  ∃ P,
    (∀ t ∈ P, t ⊆ ⇑g ⁻¹' {i}) ∧
      (↑P).PairwiseDisjoint id ∧
        (∀ t ∈ P, MeasurableSet t) ∧
          ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤ ∑ p ∈ P, ‖((μ.transpose B).restrict s) p‖ₑ + ρ
⊢ ∃ P,
    (∀ t ∈ P, t ⊆ ⇑g ⁻¹' {i}) ∧
      (↑P).PairwiseDisjoint id ∧
        (∀ t ∈ P, MeasurableSet t) ∧
          ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
            ‖i‖ₑ * (∑ p ∈ P, ‖((μ.transpose B).restrict s) p‖ₑ + ρ)

Tactic:
by

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
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : MeasurableSpace X
inst✝ : CompleteSpace G
a : X
v : F
hfm : StronglyMeasurable f
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ IsFiniteMeasure ((dirac a v).transpose B).variation

Tactic:
by

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
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : MeasurableSpace X
inst✝ : CompleteSpace G
a : X
v : F
hfm : StronglyMeasurable f
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : IsFiniteMeasure ((dirac a v).transpose B).variation
⊢ ∫ᵛ (x : X), f x ∂[B; dirac a v] = ∫ᵛ (x : X), f a ∂[B; dirac a v]

Tactic:
by

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
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : MeasurableSpace X
inst✝ : CompleteSpace G
a : X
v : F
hfm : StronglyMeasurable f
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : IsFiniteMeasure ((dirac a v).transpose B).variation
⊢ ∫ᵛ (x : X), f a ∂[B; dirac a v] = (B (f a)) v

Tactic:
by

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
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : MeasurableSpace X
inst✝ : CompleteSpace G
a : X
v : F
hfm : StronglyMeasurable f
⊢ ∫ᵛ (x : X), f x ∂[B; dirac a v] = (B (f a)) v

Tactic:
by

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
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
ι : Type u_7
hf : μ.Integrable f
l : Filter ι
s : ι → Set X
hs : Tendsto (⇑μ.variation ∘ s) l (𝓝 0)
⊢ Tendsto (fun i ↦ ∫ᵛ (x : X) in s i, f x ∂[B; μ]) l (𝓝 0)

Tactic:
by

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
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
ι : Type u_7
hf : μ.Integrable f
l : Filter ι
s : ι → Set X
hs : Tendsto (⇑μ.variation ∘ s) l (𝓝 0)
⊢ Tendsto (fun i ↦ ∫ᵛ (x : X) in s i, f x ∂[B; μ]) l (𝓝 0)

Tactic:
by
  rw [tendsto_zero_iff_norm_tendsto_zero]
  simp_rw [← coe_nnnorm, ← NNReal.coe_zero, NNReal.tendsto_coe, ← ENNReal.tendsto_coe,
    ENNReal.coe_zero]
  have : Tendsto (fun i ↦ ‖B‖ₑ * ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation) l (𝓝 (‖B‖ₑ * 0)) :=
    ENNReal.Tendsto.const_mul (tendsto_setLIntegral_zero (ne_of_lt hf.2) hs) (by simp)
  rw [mul_zero] at this
  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds this (fun i ↦ zero_le)
  intro i
  apply enorm_integral_le_lintegral_enorm.trans
  dsimp
  gcongr
  exact variation_restrict_le

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
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝² : MeasurableSpace X
inst✝¹ : MeasurableSingletonClass X
inst✝ : CompleteSpace G
a : X
v : F
⊢ IsFiniteMeasure ((dirac a v).transpose B).variation

Tactic:
by

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
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝² : MeasurableSpace X
inst✝¹ : MeasurableSingletonClass X
inst✝ : CompleteSpace G
a : X
v : F
this : IsFiniteMeasure ((dirac a v).transpose B).variation
⊢ ∫ᵛ (x : X), f x ∂[B; dirac a v] = ∫ᵛ (x : X), f a ∂[B; dirac a v]

Tactic:
by

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
I1 : ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation + ↑δ
I2 :
  ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation =
    ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i})
ρ : ℝ≥0∞
ρpos : ρ > 0
hρ : ∑ i ∈ g.range, ‖i‖ₑ * ρ ≤ ↑δ
i : E
hi : i ≠ 0
⊢ ↑‖B‖₊ ≠ ∞

Tactic:
by

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
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝² : MeasurableSpace X
inst✝¹ : MeasurableSingletonClass X
inst✝ : CompleteSpace G
a : X
v : F
this : IsFiniteMeasure ((dirac a v).transpose B).variation
⊢ ∫ᵛ (x : X), f a ∂[B; dirac a v] = (B (f a)) v

Tactic:
by

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
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝² : MeasurableSpace X
inst✝¹ : MeasurableSingletonClass X
inst✝ : CompleteSpace G
a : X
v : F
⊢ ∫ᵛ (x : X), f x ∂[B; dirac a v] = (B (f a)) v

Tactic:
by

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
I1 : ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation + ↑δ
I2 :
  ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation =
    ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i})
ρ : ℝ≥0∞
ρpos : ρ > 0
hρ : ∑ i ∈ g.range, ‖i‖ₑ * ρ ≤ ↑δ
i : E
⊢ ∃ P,
    (∀ t ∈ P, t ⊆ ⇑g ⁻¹' {i}) ∧
      (↑P).PairwiseDisjoint id ∧
        (∀ t ∈ P, MeasurableSet t) ∧
          ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
            ‖i‖ₑ * (∑ p ∈ P, ‖((μ.transpose B).restrict s) p‖ₑ + ρ)

Tactic:
by

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✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f : X → E
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : Unique X
inst✝ : CompleteSpace G
⊢ ∫ᵛ (x : X), f x ∂[B; μ] = ∫ᵛ (x : X), f default ∂[B; μ]

Tactic:
by

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✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f : X → E
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : Unique X
inst✝ : CompleteSpace G
⊢ ∫ᵛ (x : X), f default ∂[B; μ] = (B (f default)) (μ univ)

Tactic:
by

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
ι : Type u_8
f : X → E
hfi : AEStronglyMeasurable f μ.variation
F : ι → X → E
l : Filter ι
hFi : ∀ᶠ (i : ι) in l, μ.Integrable (F i)
hF : Tendsto (fun i ↦ eLpNorm (F i - f) 1 μ.variation) l (𝓝 0)
⊢ Tendsto (fun i ↦ ∫ᵛ (x : X), F i x ∂[B; μ]) l (𝓝 ∫ᵛ (x : X), f x ∂[B; μ])

Tactic:
by

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
I1 : ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation + ↑δ
I2 :
  ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation =
    ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i})
ρ : ℝ≥0∞
ρpos : ρ > 0
hρ : ∑ i ∈ g.range, ‖i‖ₑ * ρ ≤ ↑δ
P : E → Finset (Set X)
Pg : ∀ (i : E), ∀ t ∈ P i, t ⊆ ⇑g ⁻¹' {i}
Pdisj : ∀ (i : E), (↑(P i)).PairwiseDisjoint id
Pmeas : ∀ (i : E), ∀ t ∈ P i, MeasurableSet t
hP :
  ∀ (i : E),
    ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
      ‖i‖ₑ * (∑ p ∈ P i, ‖((μ.transpose B).restrict s) p‖ₑ + ρ)
⊢ ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
    ∑ i ∈ g.range, ‖i‖ₑ * (∑ p ∈ P i, ‖((μ.transpose B).restrict s) p‖ₑ + ρ)

Tactic:
by

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✝
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F✝
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F✝ →L[ℝ] G
ι : Type u_7
f : X → E
hfi : AEStronglyMeasurable f μ.variation
F : ι → X → E
l : Filter ι
hFi : ∀ᶠ (i : ι) in l, μ.Integrable (F i)
hF : Tendsto (fun i ↦ ∫⁻ (x : X), ‖F i x - f x‖ₑ ∂μ.variation) l (𝓝 0)
s : Set X
⊢ Tendsto (fun i ↦ ∫ᵛ (x : X) in s, F i x ∂[B; μ]) l (𝓝 ∫ᵛ (x : X) in s, f x ∂[B; μ])

Tactic:
by

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

Goal state before:
pp: s : Set ℕ
⊢ IsSemilinearSet s ↔ ∃ k, ∃ p > 0, ∀ x ≥ k, x ∈ s ↔ x + p ∈ s

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Add (M i)
s : Set ((i : ι) → M i)
hs : IsSlice s
i : ι
a : M i
⊢ IsSlice {x | x ∈ s ∧ x i = a}

Tactic:
by

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

Goal state before:
pp: M : Type u_1
inst✝⁴ : AddCommMonoid M
inst✝³ : PartialOrder M
inst✝² : WellQuasiOrderedLE M
inst✝¹ : IsOrderedCancelAddMonoid M
inst✝ : CanonicallyOrderedAdd M
s : Set M
hs : IsSlice s
hs' : s.Nonempty
x : M
hx : x ∈ s
⊢ ∀ {a b : M}, a ∈ {y | x + y ∈ s} → b ∈ {y | x + y ∈ s} → a + b ∈ {y | x + y ∈ s}

Tactic:
by

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

Goal state before:
pp: M : Type u_1
inst✝⁴ : AddCommMonoid M
inst✝³ : PartialOrder M
inst✝² : WellQuasiOrderedLE M
inst✝¹ : IsOrderedCancelAddMonoid M
inst✝ : CanonicallyOrderedAdd M
s : Set M
hs : IsSlice s
hs' : s.Nonempty
x : M
hx : x ∈ s
⊢ 0 ∈ {y | x + y ∈ s}

Tactic:
by

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

Goal state before:
pp: M : Type u_1
inst✝⁴ : AddCommMonoid M
inst✝³ : PartialOrder M
inst✝² : WellQuasiOrderedLE M
inst✝¹ : IsOrderedCancelAddMonoid M
inst✝ : CanonicallyOrderedAdd M
s : Set M
hs : IsSlice s
hs' : s.Nonempty
f : M → AddSubmonoid M :=
  fun x ↦ if hx : x ∈ s then { carrier := {y | x + y ∈ s}, add_mem' := ⋯, zero_mem' := ⋯ } else ⊥
⊢ ∀ x ∈ s, ∀ (y : M), y ∈ f x ↔ x + y ∈ s

Tactic:
by

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✝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
f : X → E
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : IsFiniteMeasure μ.variation
C : ℝ
h : ∀ᵐ (x : X) ∂μ.variation, ‖f x‖ ≤ C
⊢ ‖B‖ * (∫⁻ (a : X), ENNReal.ofReal ‖f a‖ ∂μ.variation).toReal ≤
    ‖B‖ * (∫⁻ (a : X), ENNReal.ofReal C ∂μ.variation).toReal

Tactic:
by

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✝
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F✝
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F✝ →L[ℝ] G
ι : Type u_7
f : X → E
hfi : AEStronglyMeasurable f μ.variation
F : ι → X → E
l : Filter ι
hFi : ∀ᶠ (i : ι) in l, μ.Integrable (F i)
hF : Tendsto (fun i ↦ eLpNorm (F i - f) 1 μ.variation) l (𝓝 0)
s : Set X
⊢ Tendsto (fun i ↦ ∫ᵛ (x : X) in s, F i x ∂[B; μ]) l (𝓝 ∫ᵛ (x : X) in s, f x ∂[B; μ])

Tactic:
by

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✝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
f : X → E
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : IsFiniteMeasure μ.variation
C : ℝ
h : ∀ᵐ (x : X) ∂μ.variation, ‖f x‖ ≤ C
hμ : ¬μ.variation = 0
this : (ae μ.variation).NeBot
⊢ 0 ≤ C

Tactic:
by

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

Goal state before:
pp: α : Type u_1
s : Set (α → ℕ)
A : Set ℕ
inst✝ : Finite α
hs : IsLinearSet s
⊢ A.Definable presburger s

Tactic:
by

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✝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
f : X → E
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : IsFiniteMeasure μ.variation
C : ℝ
h : ∀ᵐ (x : X) ∂μ.variation, ‖f x‖ ≤ C
⊢ ‖B‖ * (∫⁻ (a : X), ENNReal.ofReal C ∂μ.variation).toReal = ‖B‖ * (C * μ.variation.real univ)

Tactic:
by

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✝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
f : X → E
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : IsFiniteMeasure μ.variation
C : ℝ
h : ∀ᵐ (x : X) ∂μ.variation, ‖f x‖ ≤ C
⊢ ‖B‖ * (C * μ.variation.real univ) = C * ‖B‖ * μ.variation.real univ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
s : Set (α → ℕ)
A : Set ℕ
inst✝ : Finite α
hs : IsSemilinearSet s
⊢ A.Definable presburger s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
A : Set ℕ
inst✝ : Fintype α
i : α
⊢ ∀ (v : α → ℕ), v i = 0 + Pi.single i 1 ⬝ᵥ v

Tactic:
by

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
f : X → E
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
C : ℝ≥0∞
h : ∀ᵐ (x : X) ∂μ.variation, ‖f x‖ₑ ≤ C
⊢ ∫⁻ (a : X), C ∂μ.variation ≤ C * μ.variation univ

Tactic:
by

Goal state after: