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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s✝ : Set α
hs✝ : MeasurableSet s✝
ε✝ : ℝ≥0∞
hε : 0 < ε✝
s : Set α
hs : MeasurableSet s
h's : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ C, μ (t ∆ s) < ε
ε : ℝ≥0∞
εpos : 0 < ε
t : Set α
tC : t ∈ C
ht : μ (t ∆ s) < ε / 2
⊢ ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0 ∧ D.Nonempty

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

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

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

Tactic:
have h3 : MeasurableSet (resolventSet 𝕜 a) := (isOpen_resolventSet a).measurableSet

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

Goal state before:
pp: X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
s : Set (NullMeasurableSpace X μ)
hs : MeasurableSet s
hμs : μ.completion s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
⊢ ∃ t ∈ 𝒜, μ.completion (s ∆ t) < ENNReal.ofReal ε

Tactic:
obtain ⟨t, ht, hμst⟩ :=
      h𝒜.approx (toMeasurable μ s) (measurableSet_toMeasurable μ s) (by simpa) ε ε_pos

Goal state after:
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
s : Set (NullMeasurableSpace X μ)
hs : MeasurableSet s
hμs : μ.completion s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
t : Set X
ht : t ∈ 𝒜
hμst : μ (toMeasurable μ s ∆ t) < ENNReal.ofReal ε
⊢ ∃ t ∈ 𝒜, μ.completion (s ∆ t) < ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Measure.ResolventTransform

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : MeasurableSpace 𝕜
inst✝⁹ : CompleteSpace 𝕜
inst✝⁸ : NormedDivisionRing A
inst✝⁷ : NormedAlgebra 𝕜 A
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : OpensMeasurableSpace 𝕜
inst✝⁴ : CompleteSpace A
inst✝³ : SecondCountableTopology A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
⊢ Integrable (resolvent a) μ

Tactic:
refine ⟨by fun_prop, ?_⟩

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : MeasurableSpace 𝕜
inst✝⁹ : CompleteSpace 𝕜
inst✝⁸ : NormedDivisionRing A
inst✝⁷ : NormedAlgebra 𝕜 A
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : OpensMeasurableSpace 𝕜
inst✝⁴ : CompleteSpace A
inst✝³ : SecondCountableTopology A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
⊢ HasFiniteIntegral (resolvent a) μ
Module: Mathlib.MeasureTheory.Measure.SubFinite

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hf : Measurable f
hg : Measurable g
t : Set α := {x | f x ≤ g x}
ht : MeasurableSet t
⊢ (μ.withDensity (f - g)).restrict t = 0

Tactic:
simp only [restrict_eq_zero]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hf : Measurable f
hg : Measurable g
t : Set α := {x | f x ≤ g x}
ht : MeasurableSet t
⊢ (μ.withDensity (f - g)) t = 0
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
| (∫ (x : X), f x ∂μ).ofLp.1 = ∫ (x : X), (f x).fst ∂μ

Tactic:
1,

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hf : Integrable f μ
| (∫ (x : X), f x ∂μ).ofLp.1
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
| (∫ (x : X), f x ∂μ).ofLp.2 = ∫ (x : X), (f x).snd ∂μ

Tactic:
1,

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ q)
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : X → WithLp q (E × F)
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
hf : Integrable f μ
| (∫ (x : X), f x ∂μ).ofLp.2
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 • μ) s‖ₑ ≤ (‖c‖₊ • μ.variation) s

Tactic:
simp only [smul_apply, enorm_smul, Measure.smul_apply, Measure.nnreal_smul_coe_apply]

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 s‖ₑ ≤ ↑‖c‖₊ * μ.variation s
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case h
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)
i : ℕ
hi : i ≤ km (m + 1)
⊢ ball (D i) (u (m + 1)) ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))

Tactic:
grw [← subset_closure (s := ball (D i) (u m)), ball_subset_ball]

Goal state after:
case h
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)
i : ℕ
hi : i ≤ km (m + 1)
⊢ u (m + 1) ≤ u m
Module: Mathlib.MeasureTheory.VectorMeasure.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
a : α
this : hf.vectorMeasure (Iio a ∪ Icc a a) = Function.rightLim f a - atBot.limUnder f
⊢ hf.vectorMeasure (Iio a) = Function.leftLim f a - atBot.limUnder f

Tactic:
VectorMeasure.of_union (by simp) measurableSet_Iio
    measurableSet_Icc,

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
hf : BoundedVariationOn f univ
a : α
this : hf.vectorMeasure (Iio a) + hf.vectorMeasure (Icc a a) = Function.rightLim f a - atBot.limUnder f
⊢ hf.vectorMeasure (Iio a) = Function.leftLim f a - atBot.limUnder f
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
a : ℝ≥0∞
ha : a < μ.semivariation s
⊢ ∃ t ⊆ s, MeasurableSet t ∧ a < 2 * ‖μ t‖ₑ

Tactic:
obtain ⟨ℓ, hℓ, h'ℓ⟩ : ∃ ℓ ∈ {ℓ : StrongDual ℝ E | ‖ℓ‖ₑ ≤ 1},
    a < (μ.mapRange (ℓ : E →+ ℝ) ℓ.continuous).variation s := lt_biSup_iff.1 ha

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
a : ℝ≥0∞
ha : a < μ.semivariation s
ℓ : StrongDual ℝ E
hℓ : ℓ ∈ {ℓ | ‖ℓ‖ₑ ≤ 1}
h'ℓ : a < (μ.mapRange ↑ℓ ⋯).variation s
⊢ ∃ t ⊆ s, MeasurableSet t ∧ a < 2 * ‖μ t‖ₑ
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case h_add
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
⊢ Disjoint (support f) (support g) →
    Integrable f μ.variation →
      Integrable g μ.variation →
        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] →
            C ∫ᵛ (y : X), (f + g) y ∂[B; μ] = ∫ᵛ (y : X), (f + g) y ∂[(compL ℝ F G H) C ∘SL B; μ]

Tactic:
_

Goal state after:
case h_add
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
a✝ : Disjoint (support f) (support g)
⊢ Integrable f μ.variation →
    Integrable g μ.variation →
      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] →
          C ∫ᵛ (y : X), (f + g) y ∂[B; μ] = ∫ᵛ (y : X), (f + 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
⊢ Integrable f μ.variation →
    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:
_

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
⊢ 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]
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos.h_add
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
⊢ Disjoint (support f) (support g) →
    Integrable f μ.variation →
      Integrable g μ.variation →
        ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ] →
          ∫ᵛ (y : X), C (g y) ∂[B; μ] = ∫ᵛ (y : X), g y ∂[B ∘SL C; μ] →
            ∫ᵛ (y : X), C ((f + g) y) ∂[B; μ] = ∫ᵛ (y : X), (f + g) y ∂[B ∘SL C; μ]

Tactic:
_

Goal state after:
case pos.h_add
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
a✝ : Disjoint (support f) (support g)
⊢ Integrable f μ.variation →
    Integrable g μ.variation →
      ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ] →
        ∫ᵛ (y : X), C (g y) ∂[B; μ] = ∫ᵛ (y : X), g y ∂[B ∘SL C; μ] →
          ∫ᵛ (y : X), C ((f + g) y) ∂[B; μ] = ∫ᵛ (y : X), (f + g) y ∂[B ∘SL C; 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
⊢ Integrable f μ.variation →
    ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ] →
      ∫ᵛ (y : X), C (g y) ∂[B; μ] = ∫ᵛ (y : X), g y ∂[B ∘SL C; μ]

Tactic:
_

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
⊢ ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ] →
    ∫ᵛ (y : X), C (g y) ∂[B; μ] = ∫ᵛ (y : X), g y ∂[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✝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
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝² : CompleteSpace G
inst✝¹ : IsFiniteMeasure μ.variation
inst✝ : SFinite ν.variation
⊢ (μ.prod ν B).variation ≤ ‖B‖ₑ • μ.variation.prod ν.variation

Tactic:
apply variation_le_of_forall_enorm_le (fun s hs ↦ ?_)
  rw [prod_apply_eq_integral hs]
  simp only [Measure.smul_apply, smul_eq_mul, Measure.prod_apply hs]
  grw [enorm_integral_le_lintegral_enorm, ContinuousLinearMap.opENorm_flip,
    enorm_measure_le_variation]

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 (s ∩ r)‖ₑ + (μ.toMeasureOfLEZero sᶜ ⋯ hneg) r = ‖μ (s ∩ r)‖ₑ + ‖μ (sᶜ ∩ r)‖ₑ

Tactic:
μ.toMeasureOfLEZero_apply_eq_enorm hs.compl hneg hr

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 (s ∩ r)‖ₑ + ‖μ (sᶜ ∩ r)‖ₑ = ‖μ (s ∩ r)‖ₑ + ‖μ (sᶜ ∩ r)‖ₑ
Module: Mathlib.MeasureTheory.VectorMeasure.AddContent

Goal state before:
pp: case pos
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 : MeasurableSet s
⊢ ‖m'' s‖ₑ ≤ μ s

Tactic:
simp only [coe_mk, hs, ↓reduceIte, m'']

Goal state after:
case pos
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 : MeasurableSet s
⊢ ‖∫ᵛ (x : α), μ[s.indicator 1 | M] x ∂•m'‖ₑ ≤ μ s
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Basic

Goal state before:
pp: α : Type u_1
t : presburger.Term α
M : Type u_2
v : α → M
inst✝ : AddMonoidWithOne M
n : ℕ
⊢ Term.realize v (n • t) = n • Term.realize v t

Tactic:
induction n with simp [*, add_nsmul]

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
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
this : μ.variation ≤ C⁻¹ • (μ.transpose B).variation
⊢ (C * C⁻¹) • (μ.transpose B).variation ≤ (μ.transpose B).variation

Tactic:
mul_inv_cancel₀ hC,

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
this : μ.variation ≤ C⁻¹ • (μ.transpose B).variation
⊢ 1 • (μ.transpose B).variation ≤ (μ.transpose B).variation
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
n : ℕ
x : M
hx : x ∈ closure t
⊢ a ∈ a +ᵥ ↑(closure t)

Tactic:
nth_rw 2 [← add_zero a]

Goal state after:
case mpr.inr
M : Type u_1
inst✝ : AddCommMonoid M
a : M
t : Set M
ht : t.Finite
n : ℕ
x : M
hx : x ∈ closure t
⊢ a + 0 ∈ a +ᵥ ↑(closure t)
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:
rcases subsingleton_or_nontrivial G with h'G | h'G

Goal state after:
case inl
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
h'G : Subsingleton G
⊢ ∫ᵛ (x : X), f x ∂<•μ.toSignedMeasure = ∫ (x : X), f x ∂μ

case inr
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
h'G : Nontrivial G
⊢ ∫ᵛ (x : X), f x ∂<•μ.toSignedMeasure = ∫ (x : X), f x ∂μ
Module: Mathlib.ModelTheory.Equivalence

Goal state before:
pp: L : Language
T : L.Theory
α : Type w
n : ℕ
φ ψ : L.BoundedFormula α n
h : φ ⇔[T] ψ
⊢ ∼φ ⇔[T] ∼ψ

Tactic:
by
  simp_rw [Theory.Iff, ModelsBoundedFormula, BoundedFormula.realize_iff,
    BoundedFormula.realize_not]
  exact fun M v xs => not_congr h.realize_bd_iff

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
f : M ≃ₚ[L] N
h : f.dom = ⊤
m : M
⊢ (toEmbeddingOfEqTop h) m = ↑(f.toEquiv ⟨m, ⋯⟩)

Tactic:
rcases f with ⟨dom, cod, g⟩
  cases h
  rfl

Goal state after:
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
a₁✝ : M ↪[L] N
⊢ ∀ ⦃a₂ : M ↪[L] N⦄, (fun f ↦ f.toPartialEquiv) a₁✝ = (fun f ↦ f.toPartialEquiv) a₂ → a₁✝ = a₂

Tactic:
_

Goal state after:
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
a₁✝ a₂✝ : M ↪[L] N
⊢ (fun f ↦ f.toPartialEquiv) a₁✝ = (fun f ↦ f.toPartialEquiv) a₂✝ → a₁✝ = a₂✝