awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
s t : MeasuredSets μ
h : Disjoint (↑s : Set α) (↑t : Set α)
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ : 8 * δ = ε
s' : Set α
s'C : s' ∈ C
hs' : (μ : (a : Set α) → ℝ≥0∞) (s' ∆ (↑s : Set α)) < δ
t' : Set α
t'C : t' ∈ C
ht' : (μ : (a : Set α) → ℝ≥0∞) (t' ∆ (↑t : Set α)) < δ
It : ‖HSub.hSub (α := E) ((m : (a : Set α) → E) t' : E) (m₁ t)‖ₑ < δ
hμ' : (μ : (a : Set α) → ℝ≥0∞) (s' ∩ t') < 2 * δ
s'' : Set α := s' \ t'
s''C : s'' ∈ C
hs'' : (μ : (a : Set α) → ℝ≥0∞) (s'' ∆ (↑s : Set α)) < 3 * δ
Is : ‖HSub.hSub (α := E) ((m : (a : Set α) → E) s'' : E) (m₁ s)‖ₑ < 3 * δ
Ist :
  ‖HSub.hSub (α := E) ((m : (a : Set α) → E) (s'' ∪ t') : E)
        (m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯))‖ₑ <
    4 * δ
⊢ 4 * δ + 3 * δ + δ = 8 * δ

Tactic:
ring
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
s t : MeasuredSets μ
h : Disjoint (↑s : Set α) (↑t : Set α)
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
hδ : 8 * δ = ε
s' : Set α
s'C : s' ∈ C
hs' : (μ : (a : Set α) → ℝ≥0∞) (s' ∆ (↑s : Set α)) < δ
t' : Set α
t'C : t' ∈ C
ht' : (μ : (a : Set α) → ℝ≥0∞) (t' ∆ (↑t : Set α)) < δ
It : ‖HSub.hSub (α := E) ((m : (a : Set α) → E) t' : E) (m₁ t)‖ₑ < δ
hμ' : (μ : (a : Set α) → ℝ≥0∞) (s' ∩ t') < 2 * δ
s'' : Set α := s' \ t'
s''C : s'' ∈ C
hs'' : (μ : (a : Set α) → ℝ≥0∞) (s'' ∆ (↑s : Set α)) < 3 * δ
Is : ‖HSub.hSub (α := E) ((m : (a : Set α) → E) s'' : E) (m₁ s)‖ₑ < 3 * δ
Ist :
  ‖HSub.hSub (α := E) ((m : (a : Set α) → E) (s'' ∪ t') : E)
        (m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯))‖ₑ <
    4 * δ
⊢ 8 * δ = ε

Tactic:
hδ
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
hm₁ : ∀ (s : Set α), ‖m' s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) s
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
⊢ VectorMeasure α E

Tactic:
apply VectorMeasure.of_additive_of_le_measure m' (μ := μ)

Premises used: MeasureTheory.VectorMeasure.of_additive_of_le_measure
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
s : Set α
pos : ∀ (hs : MeasurableSet s), ‖m' s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) s
neg : ∀ (hs : ¬MeasurableSet s), ‖m' s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) s
⊢ ‖m' s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) s

Tactic:
by_cases hs : MeasurableSet s

Premises used: MeasurableSet
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
s : Set α
hs : MeasurableSet s
⊢ ‖m' s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) s

Tactic:
simpa [hs, m'] using hBound _
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
s : Set α
hs : ¬MeasurableSet s
⊢ ‖m' s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) s

Tactic:
simp [hs, m']
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hst : Disjoint s t
h'm :
  m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) (s ∪ t) ⋯) =
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s ⋯) +
      m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) t ⋯)
⊢ m' (s ∪ t) = m' s + m' t

Tactic:
simp only [hs, ht, MeasurableSet.union, ↓reduceDIte, m']

Premises used: reduceDIte, MeasurableSet.union
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hst : Disjoint s t
⊢ m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) (s ∪ t) ⋯) =
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s ⋯) +
      m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) t ⋯)

Tactic:
hAddit ⟨s, hs⟩ ⟨t, ht⟩ hst

Premises used: Subtype.mk
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hst : Disjoint s t
⊢ m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) (s ∪ t) ⋯) =
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s ⋯) +
      m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) t ⋯)

Tactic:
exact hAddit ⟨s, hs⟩ ⟨t, ht⟩ hst

Premises used: Subtype.mk
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
s : Set α
hs : ¬MeasurableSet s
⊢ m' s = 0

Tactic:
simp [m', hs]
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
m'' : VectorMeasure α E := of_additive_of_le_measure (μ := μ) m' ⋯ ⋯ ⋯
⊢ ∃ (m' : VectorMeasure α E),
    (∀ s ∈ C, (↑m' : Set α → E) s = (m : (a : Set α) → E) s) ∧
      ∀ (s : Set α), ‖(↑m' : Set α → E) s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) s

Tactic:
⟨m'', fun s hs ↦ ?_, fun s ↦ ?_⟩

Premises used: And.intro, Exists.intro
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
m'' : VectorMeasure α E := of_additive_of_le_measure (μ := μ) m' ⋯ ⋯ ⋯
refine_1 : ∀ (s : Set α) (hs : s ∈ C), (↑m'' : Set α → E) s = (m : (a : Set α) → E) s
refine_2 : ∀ (s : Set α), ‖(↑m'' : Set α → E) s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) s
⊢ ∃ (m' : VectorMeasure α E),
    (∀ s ∈ C, (↑m' : Set α → E) s = (m : (a : Set α) → E) s) ∧
      ∀ (s : Set α), ‖(↑m' : Set α → E) s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) s

Tactic:
refine ⟨m'', fun s hs ↦ ?_, fun s ↦ ?_⟩
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
m'' : VectorMeasure α E := of_additive_of_le_measure (μ := μ) m' ⋯ ⋯ ⋯
s : Set α
hs : s ∈ C
refine_1 : m' s = (m : (a : Set α) → E) s
⊢ (↑m'' : Set α → E) s = (m : (a : Set α) → E) s

Tactic:
change m' s = m s

Premises used: DFunLike.coe
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
m'' : VectorMeasure α E := of_additive_of_le_measure (μ := μ) m' ⋯ ⋯ ⋯
s : Set α
hs : s ∈ C
refine_1 : m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s ⋯) = (m : (a : Set α) → E) s
⊢ m' s = (m : (a : Set α) → E) s

Tactic:
simp only [hCmeas s hs, ↓reduceDIte, m']

Premises used: reduceDIte
-- Module: Mathlib.MeasureTheory.VectorMeasure.AddContent
Goal:
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : CompleteSpace E
μ : Measure α
inst_2 : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, enorm.{u_2} (E := E) ((m : (a : Set α) → E) s : E) ≤ (μ : (a : Set α) → ℝ≥0∞) s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, (μ : (a : Set α) → ℝ≥0∞) (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s : MeasuredSets μ | ∃ c ∈ C, (↑s : Set α) = c}
C'C : ∀ (s : MeasuredSets μ) (hs : s ∈ C'), (↑s : Set α) ∈ C
C'_dense : Dense C'
m₀ : (↑C' : Type u_1) → E := fun (x : (↑C' : Type u_1)) ↦ (m : (a : Set α) → E) (↑(↑x : MeasuredSets μ) : Set α)
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend (s := C') m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ (μ : (a : Set α) → ℝ≥0∞) (↑s : Set α)
hAddit :
  ∀ (s t : MeasuredSets μ) (h : Disjoint (↑s : Set α) (↑t : Set α)),
    m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) ((↑s : Set α) ∪ (↑t : Set α)) ⋯) = m₁ s + m₁ t
m' : (s : Set α) → E :=
  fun (s : Set α) ↦ if hs : MeasurableSet s then m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s hs) else 0
m'' : VectorMeasure α E := of_additive_of_le_measure (μ := μ) m' ⋯ ⋯ ⋯
s : Set α
hs : s ∈ C
⊢ m₁ (Subtype.mk (p := fun (s : Set α) ↦ MeasurableSet s) s ⋯) = (m : (a : Set α) → E) s

Tactic:
C'_dense.extend_eq lip.continuous ⟨⟨s, hCmeas _ hs⟩, ⟨s, hs, rfl⟩⟩

Premises used: Subtype.mk, LipschitzWith.continuous, Dense.extend_eq, And.intro, Exists.intro, rfl