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

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
μ : FiniteMeasure E
hμ : μ.mass = 1
h'μ : ∀ (n : ℕ), μ (K n)ᶜ ≤ u n
ν : ProbabilityMeasure E := ⟨↑μ, ⋯⟩
⊢ ν.toFiniteMeasure = μ

Tactic:
ext

Goal state after:
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
μ : FiniteMeasure E
hμ : μ.mass = 1
h'μ : ∀ (n : ℕ), μ (K n)ᶜ ≤ u n
ν : ProbabilityMeasure E := ⟨↑μ, ⋯⟩
s✝ : Set E
a✝ : MeasurableSet s✝
⊢ ↑ν.toFiniteMeasure s✝ = ↑μ s✝
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsZeroOneMeasure μ
inst✝¹ : StandardBorelSpace α
inst✝ : NeZero μ
this : IsProbabilityMeasure μ
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y
B : ℕ → Set α := fun n ↦ if h : μ (A n) = 1 then A n else (A n)ᶜ
mBn : MeasurableSet (⋂ n, B n)
hBn : μ (⋂ n, B n) = 1
neBn : (⋂ n, B n).Nonempty
y : α
n : ℕ
hsome : ⋯.some ∈ if μ (A n) = 1 then A n else (A n)ᶜ
hy : y ∈ if μ (A n) = 1 then A n else (A n)ᶜ
⊢ y ∈ A n ↔ neBn.some ∈ A n

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsZeroOneMeasure μ
inst✝¹ : StandardBorelSpace α
inst✝ : NeZero μ
this : IsProbabilityMeasure μ
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y
B : ℕ → Set α := ⋯
mBn : MeasurableSet (⋂ n, B n)
hBn : μ (⋂ n, B n) = 1
neBn : (⋂ n, B n).Nonempty
y : α
n : ℕ
hsome : ⋯.some ∈ if μ (A n) = 1 then A n else (A n)ᶜ
hy : y ∈ if μ (A n) = 1 then A n else (A n)ᶜ
⊢ y ∈ A n → neBn.some ∈ A n

case mpr
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsZeroOneMeasure μ
inst✝¹ : StandardBorelSpace α
inst✝ : NeZero μ
this : IsProbabilityMeasure μ
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y
B : ℕ → Set α := ⋯
mBn : MeasurableSet (⋂ n, B n)
hBn : μ (⋂ n, B n) = 1
neBn : (⋂ n, B n).Nonempty
y : α
n : ℕ
hsome : ⋯.some ∈ if μ (A n) = 1 then A n else (A n)ᶜ
hy : y ∈ if μ (A n) = 1 then A n else (A n)ᶜ
⊢ neBn.some ∈ A n → y ∈ A n
Module: Mathlib.MeasureTheory.SpecificCodomains.Pi

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p : ℝ≥0∞
ι : Type u_2
inst✝¹ : Fintype ι
E : ι → Type u_3
inst✝ : (i : ι) → NormedAddCommGroup (E i)
f : X → (i : ι) → E i
hf : ∀ (i : ι), MemLp (fun x ↦ f x i) p μ
⊢ f = ∑ i, Pi.single i ∘ fun x ↦ f x i

Tactic:
ext

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p : ℝ≥0∞
ι : Type u_2
inst✝¹ : Fintype ι
E : ι → Type u_3
inst✝ : (i : ι) → NormedAddCommGroup (E i)
f : X → (i : ι) → E i
hf : ∀ (i : ι), MemLp (fun x ↦ f x i) p μ
x✝¹ : X
x✝ : ι
⊢ f x✝¹ x✝ = (∑ i, Pi.single i ∘ fun x ↦ f x i) x✝¹ x✝
Module: Mathlib.MeasureTheory.SpecificCodomains.Pi

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p : ℝ≥0∞
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
f : X → E × F
h : MemLp (fun x ↦ (f x).1) p μ ∧ MemLp (fun x ↦ (f x).2) p μ
⊢ f = (⇑(AddMonoidHom.inl E F) ∘ fun x ↦ (f x).1) + ⇑(AddMonoidHom.inr E F) ∘ fun x ↦ (f x).2

Tactic:
ext

Goal state after:
case fst
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p : ℝ≥0∞
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
f : X → E × F
h : MemLp (fun x ↦ (f x).1) p μ ∧ MemLp (fun x ↦ (f x).2) p μ
x✝ : X
⊢ (f x✝).1 = (((⇑(AddMonoidHom.inl E F) ∘ fun x ↦ (f x).1) + ⇑(AddMonoidHom.inr E F) ∘ fun x ↦ (f x).2) x✝).1

case snd
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p : ℝ≥0∞
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
f : X → E × F
h : MemLp (fun x ↦ (f x).1) p μ ∧ MemLp (fun x ↦ (f x).2) p μ
x✝ : X
⊢ (f x✝).2 = (((⇑(AddMonoidHom.inl E F) ∘ fun x ↦ (f x).1) + ⇑(AddMonoidHom.inr E F) ∘ fun x ↦ (f x).2) x✝).2
Module: Mathlib.MeasureTheory.Order.UpperLower

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
s : Set (ι → ℝ)
inst✝ : Nonempty ι
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
⊢ s ⊆ closure s \ interior s

Tactic:
hs.interior_eq_empty,

Goal state after:
ι : Type u_1
inst✝¹ : Fintype ι
s : Set (ι → ℝ)
inst✝ : Nonempty ι
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
⊢ s ⊆ closure s \ ∅
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : AddCommMonoid V
inst✝¹ : TopologicalSpace V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
P : Finpartition ⟨s, hs⟩
⊢ ∑ p ∈ P.parts, μ ↑p = μ s

Tactic:
by
  rw [← μ.of_biUnion_finset (P.pairwiseDisjoint_apply (fun _ _ => rfl) rfl) (fun p _ => p.prop),
      ← Finset.sup_set_eq_biUnion, P.sup_parts_apply (fun _ _ => rfl) rfl]

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
m : Set α → E
hm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s
inst✝ : IsFiniteMeasure μ
h'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t
h''m : ∀ (s : Set α), ¬MeasurableSet s → m s = 0
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Function.onFun Disjoint f)
A : ∀ (n : ℕ), m (⋃ i ∈ Finset.range n, f i) = ∑ i ∈ Finset.range n, m (f i)
n : ℕ
⊢ ⋃ i, f i = (⋃ i ∈ Finset.range n, f i) ∪ ⋃ i ∈ Set.Ici n, f i

Tactic:
ext

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
m : Set α → E
hm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s
inst✝ : IsFiniteMeasure μ
h'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t
h''m : ∀ (s : Set α), ¬MeasurableSet s → m s = 0
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Function.onFun Disjoint f)
A : ∀ (n : ℕ), m (⋃ i ∈ Finset.range n, f i) = ∑ i ∈ Finset.range n, m (f i)
n : ℕ
x✝ : α
⊢ x✝ ∈ ⋃ i, f i ↔ x✝ ∈ (⋃ i ∈ Finset.range n, f i) ∪ ⋃ i ∈ Set.Ici n, f i
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
h'α : Nonempty α
m : AddContent E {s | ∃ u v, u ≤ v ∧ s = Ioc u v} := AddContent.onIoc (Function.rightLim f)
u v : α
huv : u ≤ v
⊢ ‖Function.rightLim f v - Function.rightLim f u‖ₑ ≤ hf.measureAux (Ioc u v)

Tactic:
simp only [BoundedVariationOn.measureAux, h'α, ↓reduceDIte, StieltjesFunction.measure_Ioc,
      BoundedVariationOn.stieltjesFunctionRightLim_apply]

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
h'α : Nonempty α
m : AddContent E {s | ∃ u v, u ≤ v ∧ s = Ioc u v} := AddContent.onIoc (Function.rightLim f)
u v : α
huv : u ≤ v
⊢ ‖Function.rightLim f v - Function.rightLim f u‖ₑ ≤
    ENNReal.ofReal
      (variationOnFromTo (Function.rightLim f) univ ⋯.some v - variationOnFromTo (Function.rightLim f) univ ⋯.some u)
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
h : μ.semivariation univ = ∞
t : Set X → Set X
t_meas : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → MeasurableSet (t s)
t_subs : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → t s ⊆ s
t_var : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → μ.semivariation (t s) = ∞
ht : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → 1 ≤ ‖μ (s \ t s)‖ₑ
s : ℕ → Set X := fun n ↦ t^[n] univ
hs : ∀ (n : ℕ), MeasurableSet (s n) ∧ μ.semivariation (s n) = ∞
u : ℕ → Set X := fun n ↦ s n \ s (n + 1)
hu : ∀ (n : ℕ), 1 ≤ ‖μ (u n)‖ₑ
s_anti : Antitone s
u_disj : Pairwise (Disjoint on u)
this✝ : HasSum (fun i ↦ μ (u i)) (μ (⋃ i, u i))
this : Tendsto (fun x ↦ ‖μ (u x)‖ₑ) atTop (𝓝 0)
n : ℕ
hn : ‖μ (u n)‖ₑ < 1
⊢ False

Tactic:
order [hu n]

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 : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : hα = generateFrom C
h''C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
⊢ ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ supClosure C, μ (s ∆ t) < ε

Tactic:
intro t ε ht εpos

Goal state after:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : hα = generateFrom C
h''C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
t : Set α
ε : ℝ≥0∞
ht : MeasurableSet t
εpos : 0 < ε
⊢ ∃ s ∈ supClosure C, μ (s ∆ t) < ε
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
hG : CompleteSpace G
c : H
s : Set X
hs : MeasurableSet s
hc : μ.variation s < ∞
this : IsFiniteMeasure (μ.variation.restrict s)
⊢ (fun y ↦ C (s.indicator (fun x ↦ c) y)) = s.indicator fun x ↦ C c

Tactic:
ext

Goal state after:
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
c : H
s : Set X
hs : MeasurableSet s
hc : μ.variation s < ∞
this : IsFiniteMeasure (μ.variation.restrict s)
x✝ : X
⊢ C (s.indicator (fun x ↦ c) x✝) = s.indicator (fun x ↦ C c) x✝
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 (x : X), f x ∂[B ∘SL C; μ]

Tactic:
← integral_congr_ae this,

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 (x : X), C (f x) ∂[B; μ] = ∫ᵛ (x : X), f x ∂[B ∘SL C; μ]
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
⊢ ∑' (i : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation = ‖B‖ₑ * ∫⁻ (x : X) in ⋃ i, s i, ‖f x‖ₑ ∂μ.variation

Tactic:
by
      rw [ENNReal.tsum_mul_left, lintegral_iUnion hm hd]

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
s : Set X
hs : MeasurableSet s
x : E
⊢ ‖((μ.transpose B) s) x‖ₑ ≤ (‖B‖₊ • μ.variation) s * ‖x‖ₑ

Tactic:
simp only [transpose, mapRange_apply, LinearMap.toAddMonoidHom_coe, coe_coe, flip_apply,
    Measure.smul_apply, Measure.nnreal_smul_coe_apply]

Goal state after:
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
s : Set X
hs : MeasurableSet s
x : E
⊢ ‖(B x) (μ s)‖ₑ ≤ ↑‖B‖₊ * μ.variation s * ‖x‖ₑ
Module: Mathlib.MeasureTheory.VectorMeasure.Integral

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

Tactic:
simp only [transpose, mapRange_apply, LinearMap.toAddMonoidHom_coe, coe_coe]

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

Goal state before:
pp: 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
⊢ IsSemilinearSet (⇑f '' s) ↔ IsSemilinearSet s

Tactic:
constructor

Goal state after:
case mp
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
⊢ IsSemilinearSet (⇑f '' s) → IsSemilinearSet s

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
⊢ IsSemilinearSet s → IsSemilinearSet (⇑f '' s)
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: M : Type u_1
ι : Type u_3
κ : Type u_4
inst✝ : AddCommMonoid M
s : Set (ι ⊕ κ → M)
hs : IsSemilinearSet s
x : ι → M
⊢ x ∈ {x | ∃ y, Sum.elim x y ∈ s} ↔ x ∈ ⇑(LinearMap.funLeft ℕ M Sum.inl) '' s

Tactic:
constructor

Goal state after:
case mp
M : Type u_1
ι : Type u_3
κ : Type u_4
inst✝ : AddCommMonoid M
s : Set (ι ⊕ κ → M)
hs : IsSemilinearSet s
x : ι → M
⊢ x ∈ {x | ∃ y, Sum.elim x y ∈ s} → x ∈ ⇑(LinearMap.funLeft ℕ M Sum.inl) '' s

case mpr
M : Type u_1
ι : Type u_3
κ : Type u_4
inst✝ : AddCommMonoid M
s : Set (ι ⊕ κ → M)
hs : IsSemilinearSet s
x : ι → M
⊢ x ∈ ⇑(LinearMap.funLeft ℕ M Sum.inl) '' s → x ∈ {x | ∃ y, Sum.elim x y ∈ s}
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: M : Type u_1
inst✝ : AddCommMonoid M
a : M
t : Set M
ht : t.Finite
x : M
⊢ x ∈ closure (a +ᵥ ↑(closure t)) ↔ x = 0 ∨ ∃ y ∈ closure (insert a t), a + y = x

Tactic:
constructor

Goal state after:
case mp
M : Type u_1
inst✝ : AddCommMonoid M
a : M
t : Set M
ht : t.Finite
x : M
⊢ x ∈ closure (a +ᵥ ↑(closure t)) → x = 0 ∨ ∃ y ∈ closure (insert a t), a + y = x

case mpr
M : Type u_1
inst✝ : AddCommMonoid M
a : M
t : Set M
ht : t.Finite
x : M
⊢ (x = 0 ∨ ∃ y ∈ closure (insert a t), a + y = x) → x ∈ closure (a +ᵥ ↑(closure t))
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
⊢ μ.transpose 0 = 0

Tactic:
ext

Goal state after:
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
i✝ : Set X
a✝ : MeasurableSet i✝
x✝ : E
⊢ ((μ.transpose 0) i✝) x✝ = (0 i✝) x✝
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 C : E →L[ℝ] F →L[ℝ] G
⊢ μ.transpose (B + C) = μ.transpose B + μ.transpose C

Tactic:
ext

Goal state after:
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B C : E →L[ℝ] F →L[ℝ] G
i✝ : Set X
a✝ : MeasurableSet i✝
x✝ : E
⊢ ((μ.transpose (B + C)) i✝) x✝ = ((μ.transpose B + μ.transpose C) i✝) x✝
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
⊢ (-μ).transpose B = -μ.transpose B

Tactic:
ext

Goal state after:
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
i✝ : Set X
a✝ : MeasurableSet i✝
x✝ : E
⊢ (((-μ).transpose B) i✝) x✝ = ((-μ.transpose B) i✝) x✝
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
⊢ μ.transpose (-B) = -μ.transpose B

Tactic:
ext

Goal state after:
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
i✝ : Set X
a✝ : MeasurableSet i✝
x✝ : E
⊢ ((μ.transpose (-B)) i✝) x✝ = ((-μ.transpose B) i✝) x✝
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 ν : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
⊢ (μ - ν).transpose B = μ.transpose B - ν.transpose B

Tactic:
ext

Goal state after:
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ ν : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
i✝ : Set X
a✝ : MeasurableSet i✝
x✝ : E
⊢ (((μ - ν).transpose B) i✝) x✝ = ((μ.transpose B - ν.transpose B) i✝) x✝
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 C : E →L[ℝ] F →L[ℝ] G
⊢ μ.transpose (B - C) = μ.transpose B - μ.transpose C

Tactic:
ext

Goal state after:
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B C : E →L[ℝ] F →L[ℝ] G
i✝ : Set X
a✝ : MeasurableSet i✝
x✝ : E
⊢ ((μ.transpose (B - C)) i✝) x✝ = ((μ.transpose B - μ.transpose C) i✝) x✝
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: 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
x : M
⊢ (∃ y ∈ closure ↑t, a +ᵥ y = x) ↔ x ∈ ⋃ j ∈ t', ⋃ k ∈ Finset.range (f j), (a + k • j) +ᵥ ↑(closure ↑(t.erase j))

Tactic:
constructor

Goal state after:
case mp
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
x : M
⊢ (∃ y ∈ closure ↑t, a +ᵥ y = x) → x ∈ ⋃ j ∈ t', ⋃ k ∈ Finset.range (f j), (a + k • j) +ᵥ ↑(closure ↑(t.erase j))

case mpr
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
x : M
⊢ x ∈ ⋃ j ∈ t', ⋃ k ∈ Finset.range (f j), (a + k • j) +ᵥ ↑(closure ↑(t.erase j)) → ∃ y ∈ closure ↑t, a +ᵥ y = x