mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ (OuterMeasure.ofFunction ⇑m ⋯) (t ∩ s) + (OuterMeasure.ofFunction ⇑m ⋯) (t \ s) ≤ ∑' (i : ℕ), m (f i)

Tactic:
let A : ℕ → Finset (Set α) := fun i ↦ hC.disjointOfDiff (hf i) (hC.inter_mem _ (hf i) _ hs)

Goal state after:
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 ⋯ ⋯
⊢ (OuterMeasure.ofFunction ⇑m ⋯) (t ∩ s) + (OuterMeasure.ofFunction ⇑m ⋯) (t \ s) ≤ ∑' (i : ℕ), m (f i)
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: case hs
α : Type u_1
C : Set (Set α)
s : Set α
mα : MeasurableSpace α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
hC_gen : mα = MeasurableSpace.generateFrom C
m_sigma_subadd : m.IsSigmaSubadditive
hs : s ∈ C
⊢ MeasurableSet s

Tactic:
apply MeasurableSpace.measurableSet_generateFrom hs

Goal state after:
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:
enter [1, 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
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:
enter [1, 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
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case refine_2
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
⊢ Dense D

Tactic:
refine Lp.induction p_ne_top.elim (motive := fun f ↦ f ∈ closure D) ?_ ?_ isClosed_closure

Goal state after:
case refine_2.refine_1
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
⊢ ∀ (c : E) {s : Set X} (hs : MeasurableSet s) (hμs : μ s < ∞), ↑(simpleFunc.indicatorConst p hs ⋯ c) ∈ closure D

case refine_2.refine_2
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : IsSeparable μ
inst✝ : SeparableSpace E
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : μ.MeasureDense 𝒜
𝒜₀ : Set (Set X) := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
h𝒜₀ : μ.MeasureDense 𝒜₀
count_𝒜₀ : 𝒜₀.Countable
p_ne_zero : p ≠ 0
u : Set E
countable_u : u.Countable
dense_u : Dense u
key : (n : ℕ) → (Fin n → ↑u) → (Fin n → ↑𝒜₀) → ↥(Lp E p μ) := fun n d s ↦ ∑ i, indicatorConstLp p ⋯ ⋯ ↑(d i)
D : Set ↥(Lp E p μ) := {s | ∃ n d t, s = key n d t}
⊢ ∀ ⦃f g : X → E⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    Disjoint (Function.support f) (Function.support g) →
      MemLp.toLp f hf ∈ closure D → MemLp.toLp g hg ∈ closure D → MemLp.toLp f hf + MemLp.toLp g hg ∈ closure D
Module: Mathlib.MeasureTheory.Order.UpperLower

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : s.OrdConnected
⊢ volume (frontier (↑(upperClosure s) ∩ ↑(lowerClosure s))) = 0

Tactic:
exact measure_mono_null (frontier_inter_subset _ _) <| measure_union_null
    (measure_inter_null_of_null_left _ (UpperSet.upper _).null_frontier)
    (measure_inter_null_of_null_right _ (LowerSet.lower _).null_frontier)

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

Goal state before:
pp: case insert
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
s : ι → Set X
a✝ : ι
s✝ : Finset ι
hat : a✝ ∉ s✝
IH :
  (∀ i ∈ s✝, MeasurableSet (s i)) →
    (↑s✝).Pairwise (Disjoint on s) →
      (∀ i ∈ s✝, μ.IntegrableOn f (s i)) →
        ∫ᵛ (x : X) in ⋃ i ∈ s✝, s i, f x ∂[B; μ] = ∑ i ∈ s✝, ∫ᵛ (x : X) in s i, f x ∂[B; μ]
hs : MeasurableSet (s a✝) ∧ ∀ x ∈ s✝, MeasurableSet (s x)
hf : μ.IntegrableOn f (s a✝) ∧ ∀ x ∈ s✝, μ.IntegrableOn f (s x)
h's : (↑s✝).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑s✝, a✝ ≠ b → (Disjoint on s) a✝ b ∧ (Disjoint on s) b a✝
⊢ ∫ᵛ (x : X) in s a✝, f x ∂[B; μ] + ∫ᵛ (x : X) in ⋃ x ∈ s✝, s x, f x ∂[B; μ] =
    ∑ i ∈ insert a✝ s✝, ∫ᵛ (x : X) in s i, f x ∂[B; μ]

Tactic:
Finset.sum_insert hat,

Goal state after:
case insert
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
s : ι → Set X
a✝ : ι
s✝ : Finset ι
hat : a✝ ∉ s✝
IH :
  (∀ i ∈ s✝, MeasurableSet (s i)) →
    (↑s✝).Pairwise (Disjoint on s) →
      (∀ i ∈ s✝, μ.IntegrableOn f (s i)) →
        ∫ᵛ (x : X) in ⋃ i ∈ s✝, s i, f x ∂[B; μ] = ∑ i ∈ s✝, ∫ᵛ (x : X) in s i, f x ∂[B; μ]
hs : MeasurableSet (s a✝) ∧ ∀ x ∈ s✝, MeasurableSet (s x)
hf : μ.IntegrableOn f (s a✝) ∧ ∀ x ∈ s✝, μ.IntegrableOn f (s x)
h's : (↑s✝).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑s✝, a✝ ≠ b → (Disjoint on s) a✝ b ∧ (Disjoint on s) b a✝
⊢ ∫ᵛ (x : X) in s a✝, f x ∂[B; μ] + ∫ᵛ (x : X) in ⋃ x ∈ s✝, s x, f x ∂[B; μ] =
    ∫ᵛ (x : X) in s a✝, f x ∂[B; μ] + ∑ x ∈ s✝, ∫ᵛ (x : X) in s x, f x ∂[B; μ]
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 : α
hf : BoundedVariationOn f univ
ha : ¬IsBot a
b : α
hb : b < a
u : ℕ → α
u_mono : StrictMono u
u_lim : Tendsto u atTop (𝓝 a)
u_lt_a : ∀ (n : ℕ), u n < a
A : {a} = ⋂ n, Ioc (u n) a
L1 : Tendsto (fun n ↦ hf.vectorMeasure (Ioc (u n) a)) atTop (𝓝 (hf.vectorMeasure {a}))
⊢ Tendsto (fun n ↦ hf.vectorMeasure (Ioc (u n) a)) atTop (𝓝 (Function.rightLim f a - Function.leftLim f a))

Tactic:
hf.vectorMeasure_Ioc (u_lt_a _).le

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 : α
hf : BoundedVariationOn f univ
ha : ¬IsBot a
b : α
hb : b < a
u : ℕ → α
u_mono : StrictMono u
u_lim : Tendsto u atTop (𝓝 a)
u_lt_a : ∀ (n : ℕ), u n < a
A : {a} = ⋂ n, Ioc (u n) a
L1 : Tendsto (fun n ↦ hf.vectorMeasure (Ioc (u n) a)) atTop (𝓝 (hf.vectorMeasure {a}))
⊢ Tendsto (fun n ↦ Function.rightLim f a - Function.rightLim f (u n)) atTop
    (𝓝 (Function.rightLim f a - Function.leftLim f a))
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation

Goal state before:
pp: X : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
mX : MeasurableSpace X
μ : VectorMeasure X E
s : Set X
hs : MeasurableSet s
h's : μ.semivariation s = ∞
t : Set X
ts : t ⊆ s
t_meas : MeasurableSet t
ht : 2 * ‖μ s‖ₑ + 2 < 2 * ‖μ t‖ₑ
⊢ 2 * ‖μ s‖ₑ + 2 * 1 ≤ 2 * ‖μ t‖ₑ

Tactic:
mul_one

Goal state after:
X : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
mX : MeasurableSpace X
μ : VectorMeasure X E
s : Set X
hs : MeasurableSet s
h's : μ.semivariation s = ∞
t : Set X
ts : t ⊆ s
t_meas : MeasurableSet t
ht : 2 * ‖μ s‖ₑ + 2 < 2 * ‖μ t‖ₑ
⊢ 2 * ‖μ s‖ₑ + 2 ≤ 2 * ‖μ t‖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
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
⊢ ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ]

Tactic:
by_cases hG : CompleteSpace G

Goal state after:
case pos
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
⊢ ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ]

case neg
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
⊢ ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ]
Module: Mathlib.MeasureTheory.VectorMeasure.Prod

Goal state before:
pp: case h_bound.hd
ι : Type u_1
X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
H : Type u_7
I : Type u_8
J : Type u_9
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : NormedAddCommGroup H
inst✝⁵ : NormedSpace ℝ H
inst✝⁴ : NormedAddCommGroup I
inst✝³ : NormedSpace ℝ I
inst✝² : NormedAddCommGroup J
inst✝¹ : NormedSpace ℝ J
μ✝ : VectorMeasure X E
ν✝ : VectorMeasure Y F
B✝ : E →L[ℝ] F →L[ℝ] G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : IsFiniteMeasure μ.variation
f : ℕ → Set (X × Y)
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Disjoint on f)
A :
  ∀ (a : Finset ℕ),
    ∑ y ∈ a, ∫ᵛ (x : X), ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ] = ∫ᵛ (x : X), ∑ y ∈ a, ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ]
a : Finset ℕ
x : X
⊢ (↑a).PairwiseDisjoint fun y ↦ Prod.mk x ⁻¹' f y

pp: case h_bound.hm
ι : Type u_1
X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
H : Type u_7
I : Type u_8
J : Type u_9
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : NormedAddCommGroup H
inst✝⁵ : NormedSpace ℝ H
inst✝⁴ : NormedAddCommGroup I
inst✝³ : NormedSpace ℝ I
inst✝² : NormedAddCommGroup J
inst✝¹ : NormedSpace ℝ J
μ✝ : VectorMeasure X E
ν✝ : VectorMeasure Y F
B✝ : E →L[ℝ] F →L[ℝ] G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : IsFiniteMeasure μ.variation
f : ℕ → Set (X × Y)
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Disjoint on f)
A :
  ∀ (a : Finset ℕ),
    ∑ y ∈ a, ∫ᵛ (x : X), ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ] = ∫ᵛ (x : X), ∑ y ∈ a, ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ]
a : Finset ℕ
x : X
⊢ ∀ b ∈ a, MeasurableSet (Prod.mk x ⁻¹' f b)

Tactic:
· exact fun i hi j hj hij ↦ (f_disj hij).preimage _

Goal state after:
case h_bound.hm
ι : Type u_1
X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
H : Type u_7
I : Type u_8
J : Type u_9
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : NormedAddCommGroup H
inst✝⁵ : NormedSpace ℝ H
inst✝⁴ : NormedAddCommGroup I
inst✝³ : NormedSpace ℝ I
inst✝² : NormedAddCommGroup J
inst✝¹ : NormedSpace ℝ J
μ✝ : VectorMeasure X E
ν✝ : VectorMeasure Y F
B✝ : E →L[ℝ] F →L[ℝ] G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝ : IsFiniteMeasure μ.variation
f : ℕ → Set (X × Y)
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Disjoint on f)
A :
  ∀ (a : Finset ℕ),
    ∑ y ∈ a, ∫ᵛ (x : X), ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ] = ∫ᵛ (x : X), ∑ y ∈ a, ν (Prod.mk x ⁻¹' f y) ∂[B.flip; μ]
a : Finset ℕ
x : X
⊢ ∀ b ∈ a, MeasurableSet (Prod.mk x ⁻¹' f b)
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
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
C : ℝ≥0∞
hC : ∀ᵐ (x : X) ∂μ.variation.restrict s, ‖f x‖ₑ ≤ C
hs : MeasurableSet s
⊢ ‖∫ᵛ (x : X) in s, f x ∂[B; μ]‖ₑ ≤ C * ‖B‖ₑ * μ.variation s

Tactic:
rw [← variation_restrict hs] at hC

Goal state after:
case pos
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
C : ℝ≥0∞
hC : ∀ᵐ (x : X) ∂(μ.restrict s).variation, ‖f x‖ₑ ≤ C
hs : MeasurableSet s
⊢ ‖∫ᵛ (x : X) in s, f x ∂[B; μ]‖ₑ ≤ C * ‖B‖ₑ * μ.variation s
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
C : ℝ
h : IsFiniteMeasure (μ.variation.restrict s)
hC : ∀ x ∈ s, ‖f x‖ ≤ C
⊢ ‖∫ᵛ (x : X) in s, f x ∂[B; μ]‖ ≤ C * ‖B‖ * μ.variation.real s

Tactic:
rcases eq_empty_or_nonempty s with rfl | ⟨x, hx⟩
  · simp
  by_cases hs : MeasurableSet s; swap
  · simp only [setIntegral_eq_zero_of_not_measurableSet hs, norm_zero]
    have : 0 ≤ C := le_trans (norm_nonneg _) (hC x hx)
    positivity
  apply norm_setIntegral_le_of_norm_le_const_ae
  filter_upwards [ae_restrict_mem hs] with x hx using hC x hx

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
s : ℕ → Set X
hm : ∀ (i : ℕ), 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_cases hG : CompleteSpace G

Goal state after:
case pos
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
⊢ HasSum (fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]) ∫ᵛ (x : X) in ⋃ n, s n, f x ∂[B; μ]

case neg
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
⊢ HasSum (fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]) ∫ᵛ (x : X) in ⋃ n, s n, f x ∂[B; μ]
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: M : Type u_1
N : Type u_2
F : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : FunLike F M N
inst✝ : AddMonoidHomClass F M N
f : F
a : M
t : Set M
ht : t.Finite
⊢ IsLinearSet (⇑f '' (a +ᵥ ↑(closure t)))

Tactic:
refine ⟨f a, f '' t, ht.image f, ?_⟩

Goal state after:
M : Type u_1
N : Type u_2
F : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : FunLike F M N
inst✝ : AddMonoidHomClass F M N
f : F
a : M
t : Set M
ht : t.Finite
⊢ ⇑f '' (a +ᵥ ↑(closure t)) = f a +ᵥ ↑(closure (⇑f '' t))
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: case neg
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 g : X → E
B : E →L[ℝ] F →L[ℝ] G
h : f =ᵐ[μ.variation] g
hf : ¬μ.Integrable f
this : ¬μ.Integrable g
⊢ μ.withDensity f B = μ.withDensity g B

Tactic:
simp [withDensity, hf, this]

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

Goal state before:
pp: case neg
M : Type u_1
inst✝¹ : AddCommMonoid M
inst✝ : IsCancelAdd M
a : M
t : Finset M
ih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))
t' : Finset M
ht' : t' ⊆ t
f : M → ℕ
i : M
hi : i ∈ t'
hfi : 0 < f i
heq : ∑ x ∈ t', f x • x = ∑ x ∈ t \ t', f x • x
⊢ IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))

Tactic:
convert_to IsProperSemilinearSet (⋃ j ∈ t', ⋃ k ∈ Finset.range (f j),
    (a + k • j) +ᵥ (closure (t.erase j : Set M) : Set M))

Goal state after:
M : Type u_1
inst✝¹ : AddCommMonoid M
inst✝ : IsCancelAdd M
a : M
t : Finset M
ih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))
t' : Finset M
ht' : t' ⊆ t
f : M → ℕ
i : M
hi : i ∈ t'
hfi : 0 < f i
heq : ∑ x ∈ t', f x • x = ∑ x ∈ t \ t', f x • x
⊢ a +ᵥ ↑(closure ↑t) = ⋃ j ∈ t', ⋃ k ∈ Finset.range (f j), (a + k • j) +ᵥ ↑(closure ↑(t.erase j))

case neg
M : Type u_1
inst✝¹ : AddCommMonoid M
inst✝ : IsCancelAdd M
a : M
t : Finset M
ih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))
t' : Finset M
ht' : t' ⊆ t
f : M → ℕ
i : M
hi : i ∈ t'
hfi : 0 < f i
heq : ∑ x ∈ t', f x • x = ∑ x ∈ t \ t', f x • x
⊢ IsProperSemilinearSet (⋃ j ∈ t', ⋃ k ∈ Finset.range (f j), (a + k • j) +ᵥ ↑(closure ↑(t.erase j)))
Module: Mathlib.MeasureTheory.VectorMeasure.Integral

Goal state before:
pp: case pos
X : Type u_2
E : Type u_4
F : Type u_5
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
f : X → E
μ : VectorMeasure X F
hf : μ.Integrable f
s : Set X
hs : MeasurableSet s
⊢ (μ.restrict s).Integrable f

Tactic:
simpa [VectorMeasure.Integrable, transpose_restrict, variation_restrict hs] using
      MeasureTheory.Integrable.restrict hf

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
f : X → E
c : ℝ
⊢ ∫ᵛ (x : X), f x ∂[B; c • μ] = c • ∫ᵛ (x : X), f x ∂[B; μ]

Tactic:
by_cases hG : CompleteSpace G

Goal state after:
case pos
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
f : X → E
c : ℝ
hG : CompleteSpace G
⊢ ∫ᵛ (x : X), f x ∂[B; c • μ] = c • ∫ᵛ (x : X), f 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✝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
f : X → E
c : ℝ
hG : ¬CompleteSpace G
⊢ ∫ᵛ (x : X), f x ∂[B; c • μ] = c • ∫ᵛ (x : X), f x ∂[B; μ]
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: case inr.mpr
S : Finset (Set ℕ)
hS : ∀ t ∈ S, IsProperLinearSet t
a b : ℕ
ht : LinearIndepOn ℕ id ↑{b}
hb : 0 < b
x : ℕ
hx : x ≥ a
y : ℕ
heq : a + y * b = x + b
⊢ ∃ a_1, a + a_1 * b = x

Tactic:
cases y with
          | zero => exact ⟨0, by grind⟩
          | succ y => exact ⟨y, by grind⟩

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
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
⊢ ‖B‖ * (∫⁻ (a : X), ENNReal.ofReal C ∂μ.variation).toReal = ‖B‖ * (C * μ.variation.real univ)

Tactic:
have : (ae μ.variation).NeBot := ae_neBot.mpr hμ

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
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
⊢ ‖B‖ * (∫⁻ (a : X), ENNReal.ofReal C ∂μ.variation).toReal = ‖B‖ * (C * μ.variation.real univ)