mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
i : Set α
hi : MeasurableSet i
hi₂ : v ≤[i] w
j : Set α
hj : j ⊆ i
hj₁ : MeasurableSet j
⊢ v j ≤ w j

pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
i : Set α
hi : MeasurableSet i
hi₂ : v ≤[i] w
j : Set α
hj : j ⊆ i
hj₁ : ¬MeasurableSet j
⊢ v j ≤ w j

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
i : Set α
hi : MeasurableSet i
hi₂ : v ≤[i] w
j : Set α
hj : j ⊆ i
hj₁ : MeasurableSet j
⊢ v j ≤ w j
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
i : Set α
hi : MeasurableSet i
hi₂ : v ≤[i] w
j : Set α
hj : j ⊆ i
hj₁ : ¬MeasurableSet j
⊢ v j ≤ w j

Tactic:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
i : Set α
hi : MeasurableSet i
hi₂ : v ≤[i] w
j : Set α
hj : j ⊆ i
hj₁ : ¬MeasurableSet j
⊢ v j ≤ w j
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
i : Set α
hi : MeasurableSet i
hi₂ : v ≤[i] w
j : Set α
hj : j ⊆ i
hj₁ : ¬MeasurableSet j
⊢ v j ≤ w j

Tactic:
· rw [v.not_measurable hj₁, w.not_measurable hj₁]

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

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
i : Set α
h : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → v j ≤ w j
hi : MeasurableSet i
⊢ v ≤[i] w

pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
i : Set α
h : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → v j ≤ w j
hi : ¬MeasurableSet i
⊢ v ≤[i] w

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
i : Set α
h : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → v j ≤ w j
hi : MeasurableSet i
⊢ v ≤[i] w
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
i : Set α
h : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → v j ≤ w j
hi : ¬MeasurableSet i
⊢ v ≤[i] w

Tactic:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
i : Set α
h : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → v j ≤ w j
hi : ¬MeasurableSet i
⊢ v ≤[i] w
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case pos
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h✝ : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
h₁ : ∀ l < k, 0 ≤ s (s.restrictNonposSeq i l)
l : ℕ
h : l < k
⊢ 0 ≤ s (⋃ (_ : l < k), s.restrictNonposSeq i l)

pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h✝ : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
h₁ : ∀ l < k, 0 ≤ s (s.restrictNonposSeq i l)
l : ℕ
h : ¬l < k
⊢ 0 ≤ s (⋃ (_ : l < k), s.restrictNonposSeq i l)

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h✝ : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
h₁ : ∀ l < k, 0 ≤ s (s.restrictNonposSeq i l)
l : ℕ
h : l < k
⊢ 0 ≤ s (⋃ (_ : l < k), s.restrictNonposSeq i l)
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : MeasurableSpace α
j : JordanDecomposition α
S : Set α
hS₁ : MeasurableSet S
hS₂ : j.posPart S = 0
hS₃ : j.negPart Sᶜ = 0
⊢ j.toSignedMeasure ≤[S] 0

pp: case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
j : JordanDecomposition α
S : Set α
hS₁ : MeasurableSet S
hS₂ : j.posPart S = 0
hS₃ : j.negPart Sᶜ = 0
⊢ 0 ≤[Sᶜ] j.toSignedMeasure

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝ : MeasurableSpace α
j : JordanDecomposition α
S : Set α
hS₁ : MeasurableSet S
hS₂ : j.posPart S = 0
hS₃ : j.negPart Sᶜ = 0
⊢ j.toSignedMeasure ≤[S] 0
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ∞
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ({x | ‖f x‖ < M}.indicator f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal δ
x : α
hx : ¬M ≤ ‖f x‖
⊢ f x = ({x | M ≤ ↑‖f x‖₊}.indicator f + {x | ‖f x‖ < M}.indicator f) x

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ∞
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ({x | ‖f x‖ < M}.indicator f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal δ
x : α
hx : ¬M ≤ ‖f x‖
⊢ f x = ({x | M ≤ ↑‖f x‖₊}.indicator f + {x | ‖f x‖ < M}.indicator f) x
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
j : JordanDecomposition α
S : Set α
hS₁ : MeasurableSet S
hS₂ : j.posPart S = 0
hS₃ : j.negPart Sᶜ = 0
⊢ 0 ≤[Sᶜ] j.toSignedMeasure

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
j : JordanDecomposition α
S : Set α
hS₁ : MeasurableSet S
hS₂ : j.posPart S = 0
hS₃ : j.negPart Sᶜ = 0
⊢ 0 ≤[Sᶜ] j.toSignedMeasure
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : MeasurableSpace α
s✝ s : SignedMeasure α
i : Set α := ⋯.choose
hi : MeasurableSet ⋯.choose ∧ 0 ≤[⋯.choose] s ∧ s ≤[⋯.chooseᶜ] 0
⊢ (s.toMeasureOfZeroLE i ⋯ ⋯) iᶜ = 0

pp: case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
s✝ s : SignedMeasure α
i : Set α := ⋯.choose
hi : MeasurableSet ⋯.choose ∧ 0 ≤[⋯.choose] s ∧ s ≤[⋯.chooseᶜ] 0
⊢ (s.toMeasureOfLEZero iᶜ ⋯ ⋯) iᶜᶜ = 0

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝ : MeasurableSpace α
s✝ s : SignedMeasure α
i : Set α := ⋯.choose
hi : MeasurableSet ⋯.choose ∧ 0 ≤[⋯.choose] s ∧ s ≤[⋯.chooseᶜ] 0
⊢ (s.toMeasureOfZeroLE i ⋯ ⋯) iᶜ = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
s✝ s : SignedMeasure α
i : Set α := ⋯.choose
hi : MeasurableSet ⋯.choose ∧ 0 ≤[⋯.choose] s ∧ s ≤[⋯.chooseᶜ] 0
⊢ (s.toMeasureOfLEZero iᶜ ⋯ ⋯) iᶜᶜ = 0

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
s✝ s : SignedMeasure α
i : Set α := ⋯.choose
hi : MeasurableSet ⋯.choose ∧ 0 ≤[⋯.choose] s ∧ s ≤[⋯.chooseᶜ] 0
⊢ (s.toMeasureOfLEZero iᶜ ⋯ ⋯) iᶜᶜ = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h✝ : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
h₁ : ∀ l < k, 0 ≤ s (s.restrictNonposSeq i l)
l : ℕ
h : ¬l < k
⊢ 0 ≤ s (⋃ (_ : l < k), s.restrictNonposSeq i l)

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h✝ : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
h₁ : ∀ l < k, 0 ≤ s (s.restrictNonposSeq i l)
l : ℕ
h : ¬l < k
⊢ 0 ≤ s (⋃ (_ : l < k), s.restrictNonposSeq i l)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ ∑' (i : ℕ), v (a ∩ disjointed f i) ≤ ∑' (i : ℕ), w (a ∩ disjointed f i)

pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ ∀ (i : ℕ), MeasurableSet (a ∩ disjointed f i)

pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ ∀ (i : ℕ), MeasurableSet (a ∩ disjointed f i)

Tactic:
·

Goal state after:
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ ∑' (i : ℕ), v (a ∩ disjointed f i) ≤ ∑' (i : ℕ), w (a ∩ disjointed f i)
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case neg.hm
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
⊢ ∀ (i_1 : ℕ), MeasurableSet (⋃ (_ : i_1 < k), s.restrictNonposSeq i i_1)

pp: case neg.hd
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
⊢ Pairwise (Function.onFun Disjoint fun l ↦ ⋃ (_ : l < k), s.restrictNonposSeq i l)

pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
⊢ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l ⊆ i

Tactic:
·

Goal state after:
case neg.hm
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
⊢ ∀ (i_1 : ℕ), MeasurableSet (⋃ (_ : i_1 < k), s.restrictNonposSeq i i_1)
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case neg.hd
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
⊢ Pairwise (Function.onFun Disjoint fun l ↦ ⋃ (_ : l < k), s.restrictNonposSeq i l)

pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
⊢ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l ⊆ i

Tactic:
·

Goal state after:
case neg.hd
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
⊢ Pairwise (Function.onFun Disjoint fun l ↦ ⋃ (_ : l < k), s.restrictNonposSeq i l)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case refine_1
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
n : ℕ
⊢ MeasurableSet (a ∩ disjointed f n)

pp: case refine_2
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
n : ℕ
⊢ a ∩ disjointed f n ⊆ f n

pp: case refine_3
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ Summable fun i ↦ v (a ∩ disjointed f i)

pp: case refine_4
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ Summable fun i ↦ w (a ∩ disjointed f i)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
n : ℕ
⊢ MeasurableSet (a ∩ disjointed f n)
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
⊢ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l ⊆ i

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l)
⊢ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l ⊆ i
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
n : ℕ
⊢ a ∩ disjointed f n ⊆ f n

pp: case refine_3
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ Summable fun i ↦ v (a ∩ disjointed f i)

pp: case refine_4
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ Summable fun i ↦ w (a ∩ disjointed f i)

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
n : ℕ
⊢ a ∩ disjointed f n ⊆ f n
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case refine_3
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ Summable fun i ↦ v (a ∩ disjointed f i)

pp: case refine_4
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ Summable fun i ↦ w (a ∩ disjointed f i)

Tactic:
·

Goal state after:
case refine_3
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ Summable fun i ↦ v (a ∩ disjointed f i)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case refine_3.refine_1
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
n : ℕ
⊢ MeasurableSet (a ∩ disjointed f n)

pp: case refine_3.refine_2
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
⊢ Pairwise (Disjoint on fun i ↦ a ∩ disjointed f i)

Tactic:
·

Goal state after:
case refine_3.refine_1
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩ disjointed f n = a
ha₄ : Pairwise (Disjoint on fun n ↦ a ∩ disjointed f n)
n : ℕ
⊢ MeasurableSet (a ∩ disjointed f n)