Theorem: MeasureTheory.Measure.rnDeriv_add_right_of_mutuallySingular'
File: Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean

tactic: have h_ac : ν ≪ ν + ν' := Measure.AbsolutelyContinuous.rfl.add_right _
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
⊢ μ.rnDeriv (ν + ν') =ᶠ[ae ν] μ.rnDeriv ν
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
⊢ μ.rnDeriv (ν + ν') =ᶠ[ae ν] μ.rnDeriv ν

tactic: rw [haveLebesgueDecomposition_add μ ν]
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
⊢ μ.rnDeriv (ν + ν') =ᶠ[ae ν] μ.rnDeriv ν
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
⊢ (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν

tactic: have h₁ := rnDeriv_add' (μ.singularPart ν) (ν.withDensity (μ.rnDeriv ν)) (ν + ν')
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
⊢ (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
⊢ (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν

tactic: have h₂ := rnDeriv_add' (μ.singularPart ν) (ν.withDensity (μ.rnDeriv ν)) ν
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
⊢ (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
⊢ (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν

tactic: refine (Filter.EventuallyEq.trans (h_ac.ae_le h₁) ?_).trans h₂.symm
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
⊢ (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
⊢ (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν

tactic: have h₃ := rnDeriv_add_right_of_absolutelyContinuous_of_mutuallySingular
  (withDensity_absolutelyContinuous ν (μ.rnDeriv ν)) hνν'
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
⊢ (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
⊢ (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν

tactic: have h₄ : (μ.singularPart ν).rnDeriv (ν + ν') =ᵐ[ν] 0 := by
  refine h_ac.ae_eq ?_
  simp only [rnDeriv_eq_zero, MutuallySingular.add_right_iff]
  exact ⟨mutuallySingular_singularPart μ ν, hμν'.singularPart ν⟩
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
⊢ (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₄ : (μ.singularPart ν).rnDeriv (ν + ν') =ᶠ[ae ν] 0
⊢ (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν

tactic: have h₅ : (μ.singularPart ν).rnDeriv ν =ᵐ[ν] 0 := rnDeriv_singularPart μ ν
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₄ : (μ.singularPart ν).rnDeriv (ν + ν') =ᶠ[ae ν] 0
⊢ (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₄ : (μ.singularPart ν).rnDeriv (ν + ν') =ᶠ[ae ν] 0
h₅ : (μ.singularPart ν).rnDeriv ν =ᶠ[ae ν] 0
⊢ (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν

tactic: filter_upwards [h₃, h₄, h₅] with x hx₃ hx₄ hx₅
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₄ : (μ.singularPart ν).rnDeriv (ν + ν') =ᶠ[ae ν] 0
h₅ : (μ.singularPart ν).rnDeriv ν =ᶠ[ae ν] 0
⊢ (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
state_after: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₄ : (μ.singularPart ν).rnDeriv (ν + ν') =ᶠ[ae ν] 0
h₅ : (μ.singularPart ν).rnDeriv ν =ᶠ[ae ν] 0
x : α
hx₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') x = (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν x
hx₄ : (μ.singularPart ν).rnDeriv (ν + ν') x = 0 x
hx₅ : (μ.singularPart ν).rnDeriv ν x = 0 x
⊢ ((μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')) x =
    ((μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν) x

tactic: simp only [Pi.add_apply]
state_before: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₄ : (μ.singularPart ν).rnDeriv (ν + ν') =ᶠ[ae ν] 0
h₅ : (μ.singularPart ν).rnDeriv ν =ᶠ[ae ν] 0
x : α
hx₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') x = (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν x
hx₄ : (μ.singularPart ν).rnDeriv (ν + ν') x = 0 x
hx₅ : (μ.singularPart ν).rnDeriv ν x = 0 x
⊢ ((μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')) x =
    ((μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν) x
state_after: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₄ : (μ.singularPart ν).rnDeriv (ν + ν') =ᶠ[ae ν] 0
h₅ : (μ.singularPart ν).rnDeriv ν =ᶠ[ae ν] 0
x : α
hx₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') x = (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν x
hx₄ : (μ.singularPart ν).rnDeriv (ν + ν') x = 0 x
hx₅ : (μ.singularPart ν).rnDeriv ν x = 0 x
⊢ (μ.singularPart ν).rnDeriv (ν + ν') x + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') x =
    (μ.singularPart ν).rnDeriv ν x + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν x

tactic: rw [hx₃, hx₄, hx₅]
state_before: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₄ : (μ.singularPart ν).rnDeriv (ν + ν') =ᶠ[ae ν] 0
h₅ : (μ.singularPart ν).rnDeriv ν =ᶠ[ae ν] 0
x : α
hx₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') x = (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν x
hx₄ : (μ.singularPart ν).rnDeriv (ν + ν') x = 0 x
hx₅ : (μ.singularPart ν).rnDeriv ν x = 0 x
⊢ (μ.singularPart ν).rnDeriv (ν + ν') x + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') x =
    (μ.singularPart ν).rnDeriv ν x + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν x
state_after: no goals

tactic: refine h_ac.ae_eq ?_
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
⊢ (μ.singularPart ν).rnDeriv (ν + ν') =ᶠ[ae ν] 0
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
⊢ (μ.singularPart ν).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')] 0

tactic: simp only [rnDeriv_eq_zero, MutuallySingular.add_right_iff]
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
⊢ (μ.singularPart ν).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')] 0
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
⊢ μ.singularPart ν ⟂ₘ ν ∧ μ.singularPart ν ⟂ₘ ν'

tactic: exact ⟨mutuallySingular_singularPart μ ν, hμν'.singularPart ν⟩
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
    (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν]
    (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
⊢ μ.singularPart ν ⟂ₘ ν ∧ μ.singularPart ν ⟂ₘ ν'
state_after: no goals