-- Mathlib/MeasureTheory/VectorMeasure/Basic.lean
theorem mk (s : Set α) (hs : MeasurableSet s) (h₁ : ∀ t ⊆ s, MeasurableSet t → v t = 0)
    (h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → w t = 0) : v ⟂ᵥ w

Proof state:
case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ s
ht : ¬MeasurableSet t
⊢ ↑v t = 0

Tactic:
exact not_measurable v ht

Result:
no goals

Proof state:
case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ sᶜ
ht : ¬MeasurableSet t
⊢ ↑w t = 0

Tactic:
exact not_measurable w ht

Result:
no goals

Proof state:
case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ sᶜ
ht : MeasurableSet t
⊢ ↑w t = 0

Tactic:
exact h₂ t hst ht

Result:
no goals

Proof state:
case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ s
ht : MeasurableSet t
⊢ ↑v t = 0

Tactic:
exact h₁ t hst ht

Result:
no goals

Proof state:
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
⊢ v ⟂ᵥ w

Tactic:
refine ⟨s, hs, fun t hst => ?_, fun t hst => ?_⟩ <;> by_cases ht : MeasurableSet t

Result:
case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ s
ht : MeasurableSet t
⊢ ↑v t = 0

case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ s
ht : ¬MeasurableSet t
⊢ ↑v t = 0

case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ sᶜ
ht : MeasurableSet t
⊢ ↑w t = 0

case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ sᶜ
ht : ¬MeasurableSet t
⊢ ↑w t = 0