Theorem: MeasureTheory.Measure.AbsolutelyContinuous.add_left_iff
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

tactic: refine ⟨fun h ↦ ?_, fun h ↦ (h.1.add h.2).trans ?_⟩
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
⊢ μ₁ + μ₂ ≪ ν ↔ μ₁ ≪ ν ∧ μ₂ ≪ ν
state_after: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
⊢ μ₁ ≪ ν ∧ μ₂ ≪ ν

case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ ≪ ν ∧ μ₂ ≪ ν
⊢ ν + ν ≪ ν

tactic: have : ∀ s, ν s = 0 → μ₁ s = 0 ∧ μ₂ s = 0 := by intro s hs0; simpa using h hs0
state_before: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
⊢ μ₁ ≪ ν ∧ μ₂ ≪ ν
state_after: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
this : ∀ (s : Set α), ν s = 0 → μ₁ s = 0 ∧ μ₂ s = 0
⊢ μ₁ ≪ ν ∧ μ₂ ≪ ν

tactic: exact ⟨fun s hs0 ↦ (this s hs0).1, fun s hs0 ↦ (this s hs0).2⟩
state_before: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
this : ∀ (s : Set α), ν s = 0 → μ₁ s = 0 ∧ μ₂ s = 0
⊢ μ₁ ≪ ν ∧ μ₂ ≪ ν
state_after: no goals

tactic: intro s hs0
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
⊢ ∀ (s : Set α), ν s = 0 → μ₁ s = 0 ∧ μ₂ s = 0
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
s : Set α
hs0 : ν s = 0
⊢ μ₁ s = 0 ∧ μ₂ s = 0

tactic: simpa using h hs0
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
s : Set α
hs0 : ν s = 0
⊢ μ₁ s = 0 ∧ μ₂ s = 0
state_after: no goals

tactic: have : ν + ν = 2 • ν := by ext; simp [two_mul]
state_before: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ ≪ ν ∧ μ₂ ≪ ν
⊢ ν + ν ≪ ν
state_after: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ ≪ ν ∧ μ₂ ≪ ν
this : ν + ν = 2 • ν
⊢ ν + ν ≪ ν

tactic: rw [this]
state_before: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ ≪ ν ∧ μ₂ ≪ ν
this : ν + ν = 2 • ν
⊢ ν + ν ≪ ν
state_after: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ ≪ ν ∧ μ₂ ≪ ν
this : ν + ν = 2 • ν
⊢ 2 • ν ≪ ν

tactic: exact AbsolutelyContinuous.rfl.smul 2
state_before: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ ≪ ν ∧ μ₂ ≪ ν
this : ν + ν = 2 • ν
⊢ 2 • ν ≪ ν
state_after: no goals

tactic: ext
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ ≪ ν ∧ μ₂ ≪ ν
⊢ ν + ν = 2 • ν
state_after: case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ ≪ ν ∧ μ₂ ≪ ν
s✝ : Set α
a✝ : MeasurableSet s✝
⊢ (ν + ν) s✝ = (2 • ν) s✝

tactic: simp [two_mul]
state_before: case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁✝ μ₂✝ μ₃ ν✝ ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
μ₁ μ₂ ν : Measure α
h : μ₁ ≪ ν ∧ μ₂ ≪ ν
s✝ : Set α
a✝ : MeasurableSet s✝
⊢ (ν + ν) s✝ = (2 • ν) s✝
state_after: no goals