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

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝³ : PseudoMetricSpace 𝓧
inst✝² : OpensMeasurableSpace 𝓧
inst✝¹ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst✝ : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), ⋃ i, ball (D i) (u m) = univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
km : ℕ → ℕ
bigK : Set 𝓧 :=
  ⋂ m, ⋃ i, ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))
μ : ProbabilityMeasure 𝓧
hs : μ ∈ S
m : ℕ
hbound : 1 ≤ ↑(μ (⋃ i, ⋃ (_ : i ≤ km (m + 1)), ball (D i) (u (m + 1)))) + ε * ↑2 ^ (-1 + -↑m)
⊢ 1 ≤
    ↑(μ
          (⋃ i,
            ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m)))) +
      ε * ↑2 ^ (-1 + -↑m)

Tactic:
grw [hbound]

Goal state after:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝³ : PseudoMetricSpace 𝓧
inst✝² : OpensMeasurableSpace 𝓧
inst✝¹ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst✝ : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), ⋃ i, ball (D i) (u m) = univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
km : ℕ → ℕ
bigK : Set 𝓧 :=
  ⋂ m, ⋃ i, ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))
μ : ProbabilityMeasure 𝓧
hs : μ ∈ S
m : ℕ
hbound : 1 ≤ ↑(μ (⋃ i, ⋃ (_ : i ≤ km (m + 1)), ball (D i) (u (m + 1)))) + ε * ↑2 ^ (-1 + -↑m)
⊢ ↑(μ (⋃ i, ⋃ (_ : i ≤ km (m + 1)), ball (D i) (u (m + 1)))) + ε * ↑2 ^ (-1 + -↑m) ≤
    ↑(μ
          (⋃ i,
            ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m)))) +
      ε * ↑2 ^ (-1 + -↑m)
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
hst : Disjoint s t
hs : MeasurableSet s
ht : MeasurableSet t
hfs : μ.IntegrableOn f s
hft : μ.IntegrableOn f t
⊢ ∫ᵛ (x : X) in s ∪ t, f x ∂[B; μ] = ∫ᵛ (x : X), f x ∂[B; μ.restrict s + μ.restrict t]

Tactic:
μ.restrict_union hst hs ht

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
hst : Disjoint s t
hs : MeasurableSet s
ht : MeasurableSet t
hfs : μ.IntegrableOn f s
hft : μ.IntegrableOn f t
⊢ ∫ᵛ (x : X), f x ∂[B; μ.restrict s + μ.restrict t] = ∫ᵛ (x : X), f x ∂[B; μ.restrict s + μ.restrict t]
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : VectorMeasure X ℝ≥0∞
⊢ μ.variation = μ.ennrealToMeasure

Tactic:
by
  ext _ hs
  simp [preVariationFun_apply_of_ennreal, variation_apply, preVariation_apply,
    ennrealPreVariation_apply, ennrealToMeasure_apply hs]

Goal state after:
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 : 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
⊢ ∃ m', (∀ s ∈ C, m' s = m s) ∧ ∀ (s : Set α), ‖m' s‖ₑ ≤ μ s

Tactic:
have m₁_lip : LipschitzWith 1 m₁ := C'_dense.lipschitzWith_extend lip

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 : 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
⊢ ∃ m', (∀ s ∈ C, m' s = m s) ∧ ∀ (s : Set α), ‖m' s‖ₑ ≤ μ s
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
a b : α
hf : BoundedVariationOn f univ
h : a < b
this : hf.vectorMeasure (Ioc a b) = Function.rightLim f b - Function.rightLim f a
⊢ hf.vectorMeasure (Ioo a b) = Function.leftLim f b - Function.rightLim f a

Tactic:
rw [← Ioo_union_Icc_eq_Ioc h le_rfl, VectorMeasure.of_union (by simp) measurableSet_Ioo
    measurableSet_Icc, hf.vectorMeasure_Icc le_rfl] at this

Goal state after:
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
a b : α
hf : BoundedVariationOn f univ
h : a < b
this :
  hf.vectorMeasure (Ioo a b) + (Function.rightLim f b - Function.leftLim f b) =
    Function.rightLim f b - Function.rightLim f a
⊢ hf.vectorMeasure (Ioo a b) = Function.leftLim f b - Function.rightLim f a
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup G
f : X → E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
inst✝³ : MeasurableSpace X
inst✝² : MeasurableSingletonClass X
inst✝¹ : CompleteSpace G
a : X
v : F
s : Set X
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
h✝ : a ∈ s
⊢ ∫ᵛ (x : X), f x ∂[B; dirac a v] = (B (f a)) v

pp: case neg
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup G
f : X → E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
inst✝³ : MeasurableSpace X
inst✝² : MeasurableSingletonClass X
inst✝¹ : CompleteSpace G
a : X
v : F
s : Set X
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
h✝ : a ∉ s
⊢ ∫ᵛ (x : X), f x ∂[B; 0] = 0

Tactic:
· exact integral_dirac

Goal state after:
case neg
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup G
f : X → E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
inst✝³ : MeasurableSpace X
inst✝² : MeasurableSingletonClass X
inst✝¹ : CompleteSpace G
a : X
v : F
s : Set X
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
h✝ : a ∉ s
⊢ ∫ᵛ (x : X), f x ∂[B; 0] = 0
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

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

Tactic:
← integral_congr_ae hfg,

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

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

Tactic:
← integral_congr_ae hfg,

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

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

Tactic:
← integral_congr_ae hfg,

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

Goal state before:
pp: X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
mY : MeasurableSpace Y
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 E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
ρ : VectorMeasure (X × Y) G
hρ : ∀ (s : Set X) (t : Set Y), MeasurableSet s → MeasurableSet t → ρ (s ×ˢ t) = (B (μ s)) (ν t)
this : μ.HasProd ν B
⊢ (μ.prod ν B) univ = ρ univ

Tactic:
rw [← univ_prod_univ, hρ _ _ MeasurableSet.univ MeasurableSet.univ, prod_apply]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.SignedMeasure

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
r : Set X
hr : MeasurableSet r
s : Set X
hs : MeasurableSet s
hpos : 0 ≤[s] μ
hneg : μ ≤[sᶜ] 0
hposPart : μ.toJordanDecomposition.posPart = μ.toMeasureOfZeroLE s hs hpos
hnegPart : μ.toJordanDecomposition.negPart = μ.toMeasureOfLEZero sᶜ ⋯ hneg
⊢ (μ.toJordanDecomposition.posPart + μ.toJordanDecomposition.negPart) r = ‖μ (s ∩ r)‖ₑ + ‖μ (sᶜ ∩ r)‖ₑ

Tactic:
Measure.add_apply,

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
r : Set X
hr : MeasurableSet r
s : Set X
hs : MeasurableSet s
hpos : 0 ≤[s] μ
hneg : μ ≤[sᶜ] 0
hposPart : μ.toJordanDecomposition.posPart = μ.toMeasureOfZeroLE s hs hpos
hnegPart : μ.toJordanDecomposition.negPart = μ.toMeasureOfLEZero sᶜ ⋯ hneg
⊢ μ.toJordanDecomposition.posPart r + μ.toJordanDecomposition.negPart r = ‖μ (s ∩ r)‖ₑ + ‖μ (sᶜ ∩ r)‖ₑ
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
s : ℕ → Set X
hm : ∀ (i : ℕ), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hG : CompleteSpace G
⊢ ‖B‖ₑ * ∑' (i : ℕ), ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation = ‖B‖ₑ * ∫⁻ (x : X) in ⋃ i, s i, ‖f x‖ₑ ∂μ.variation

Tactic:
lintegral_iUnion hm hd

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
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
s : ℕ → Set X
hm : ∀ (i : ℕ), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hG : CompleteSpace G
⊢ ‖B‖ₑ * ∑' (i : ℕ), ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation = ‖B‖ₑ * ∑' (i : ℕ), ∫⁻ (a : X) in s i, ‖f a‖ₑ ∂μ.variation
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: case mpr
M : Type u_1
N : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
s : Set M
F : Type u_6
inst✝¹ : EquivLike F M N
inst✝ : AddEquivClass F M N
f : F
h : IsSemilinearSet s
⊢ IsSemilinearSet (⇑f '' s)

Tactic:
exact h.image f

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

Goal state before:
pp: 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 - g a + g a‖ₑ ≤ ‖f a - g a‖ₑ + ‖g a‖ₑ

Tactic:
exact enorm_add_le (f a - g a) (g a)

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

Goal state before:
pp: case neg
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
β : Type u_9
inst✝ : MeasurableSpace β
φ : X → β
hφ : MeasurableEmbedding φ
f : β → E
hfm : ¬AEStronglyMeasurable f (Measure.map φ μ.variation)
⊢ ∫ᵛ (y : β), f y ∂[B; μ.map φ] = ∫ᵛ (x : X), f (φ x) ∂[B; μ]

Tactic:
integral_non_aestronglyMeasurable,

Goal state after:
case neg
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
β : Type u_9
inst✝ : MeasurableSpace β
φ : X → β
hφ : MeasurableEmbedding φ
f : β → E
hfm : ¬AEStronglyMeasurable f (Measure.map φ μ.variation)
⊢ 0 = ∫ᵛ (x : X), f (φ x) ∂[B; μ]

case neg
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
β : Type u_9
inst✝ : MeasurableSpace β
φ : X → β
hφ : MeasurableEmbedding φ
f : β → E
hfm : ¬AEStronglyMeasurable f (Measure.map φ μ.variation)
⊢ ¬AEStronglyMeasurable f (μ.map φ).variation
Module: Mathlib.ModelTheory.Equivalence

Goal state before:
pp: L : Language
T : L.Theory
α : Type w
n : ℕ
φ ψ : L.BoundedFormula α n
M : T.ModelType
v : α → ↑M
xs : Fin n → ↑M
⊢ (ψ ⟹ φ ⊔ ψ).Realize v xs

Tactic:
by
  simp only [BoundedFormula.realize_imp, BoundedFormula.realize_sup]
  exact Or.inr

Goal state after:
Module: Mathlib.ModelTheory.Equivalence

Goal state before:
pp: L : Language
T : L.Theory
α : Type w
n : ℕ
φ ψ φ' ψ' : L.BoundedFormula α n
h : φ ⇔[T] ψ
h' : φ' ⇔[T] ψ'
⊢ φ ⟹ φ' ⇔[T] ψ ⟹ ψ'

Tactic:
simp_rw [Theory.Iff, ModelsBoundedFormula, BoundedFormula.realize_iff,
    BoundedFormula.realize_imp]
  exact fun M v xs => imp_congr h.realize_bd_iff h'.realize_bd_iff

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝³ : Preorder ι
G : ι → Type w
inst✝² : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝¹ : IsDirectedOrder ι
inst✝ : DirectedSystem G fun i j h ↦ ⇑(f i j h)
n : ℕ
R : L.Relations n
x : Fin n → Σˣ f
i j : ι
hi : i ∈ upperBounds (range (Sigma.fst ∘ x))
hj : j ∈ upperBounds (range (Sigma.fst ∘ x))
⊢ RelMap R (unify f x i hi) = RelMap R (unify f x j hj)

Tactic:
by
  obtain ⟨k, ik, jk⟩ := directed_of (· ≤ ·) i j
  rw [← (f i k ik).map_rel, comp_unify, ← (f j k jk).map_rel, comp_unify]

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

Goal state before:
pp: ι : Type u_3
s : Set (ι → ℕ)
hs : LinearIndepOn ℕ id s
t : Finset (ι → ℕ)
f : (ι → ℕ) → ℤ
ht : ↑t ⊆ s
hf : ∀ i ∉ t, f i = 0
heq : ∑ i ∈ t, f i • toRatVec i = 0
i : ι → ℕ
hi : i ∈ t
⊢ f i = 0

Tactic:
rw [linearIndepOn_iff_linearIndepOn_finset] at hs

Goal state after:
ι : Type u_3
s : Set (ι → ℕ)
hs : ∀ (t : Finset (ι → ℕ)), ↑t ⊆ s → LinearIndepOn ℕ id ↑t
t : Finset (ι → ℕ)
f : (ι → ℕ) → ℤ
ht : ↑t ⊆ s
hf : ∀ i ∉ t, f i = 0
heq : ∑ i ∈ t, f i • toRatVec i = 0
i : ι → ℕ
hi : i ∈ t
⊢ f i = 0
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: ι : Type u_3
s : Set (ι → ℕ)
hs : IsProperLinearSet s
inst✝ : Finite ι
x : ι → ℕ
x✝ : ↑hs.basisSet
| (↑⌊(hs.basis.repr (toRatVec x - toRatVec hs.base)) x✝⌋.toNat -
      ↑(-⌊(hs.basis.repr (toRatVec x - toRatVec hs.base)) x✝⌋).toNat) •
    toRatVec ↑x✝

Tactic:
Int.toNat_sub_toNat_neg,

Goal state after:
ι : Type u_3
s : Set (ι → ℕ)
hs : IsProperLinearSet s
inst✝ : Finite ι
x : ι → ℕ
x✝ : ↑hs.basisSet
| ⌊(hs.basis.repr (toRatVec x - toRatVec hs.base)) x✝⌋ • toRatVec ↑x✝
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirectedOrder ι
inst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝¹ : Nonempty ι
P : Type u₁
inst✝ : L.Structure P
g : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
n✝ : ℕ
F : L.Functions n✝
i : ι
y : Fin n✝ → G i
⊢ Quotient.lift (fun x ↦ (g x.fst) x.snd) ⋯ ⟦Structure.Sigma.mk f i (funMap F y)⟧ =
    funMap F ((Quotient.lift (fun x ↦ (g x.fst) x.snd) ⋯ ∘ Quotient.mk (setoid G f)) ∘ Structure.Sigma.mk f i ∘ y)

Tactic:
Quotient.lift_comp_mk

Goal state after:
L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirectedOrder ι
inst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝¹ : Nonempty ι
P : Type u₁
inst✝ : L.Structure P
g : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
n✝ : ℕ
F : L.Functions n✝
i : ι
y : Fin n✝ → G i
⊢ Quotient.lift (fun x ↦ (g x.fst) x.snd) ⋯ ⟦Structure.Sigma.mk f i (funMap F y)⟧ =
    funMap F ((fun x ↦ (g x.fst) x.snd) ∘ Structure.Sigma.mk f i ∘ y)
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirectedOrder ι
inst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝¹ : Nonempty ι
P : Type u₁
inst✝ : L.Structure P
g : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
n✝ : ℕ
R : L.Relations n✝
i : ι
y : Fin n✝ → G i
⊢ RelMap R ((Quotient.lift (fun x ↦ (g x.fst) x.snd) ⋯ ∘ Quotient.mk (setoid G f)) ∘ Structure.Sigma.mk f i ∘ y) ↔
    RelMap R (⇑(g i) ∘ y)

Tactic:
Quotient.lift_comp_mk

Goal state after:
L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirectedOrder ι
inst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝¹ : Nonempty ι
P : Type u₁
inst✝ : L.Structure P
g : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
n✝ : ℕ
R : L.Relations n✝
i : ι
y : Fin n✝ → G i
⊢ RelMap R ((fun x ↦ (g x.fst) x.snd) ∘ Structure.Sigma.mk f i ∘ y) ↔ RelMap R (⇑(g i) ∘ y)
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirectedOrder ι
inst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝¹ : Nonempty ι
P : Type u₁
inst✝ : L.Structure P
g : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
⊢ ⨆ i, Substructure.map (lift L ι G f g Hg).toHom (of L ι G f i).toHom.range = ⨆ i, Substructure.map (g i).toHom ⊤

Tactic:
simp_rw [Hom.range_eq_map, Substructure.map_map]

Goal state after:
L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirectedOrder ι
inst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)
inst✝¹ : Nonempty ι
P : Type u₁
inst✝ : L.Structure P
g : (i : ι) → G i ↪[L] P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x
⊢ ⨆ i, Substructure.map ((lift L ι G f g Hg).toHom.comp (of L ι G f i).toHom) ⊤ = ⨆ i, Substructure.map (g i).toHom ⊤