-- Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
theorem eq_withDensity_rnDeriv {s : Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (hs : s ⟂ₘ ν)
    (hadd : μ = s + ν.withDensity f) : ν.withDensity f = ν.withDensity (μ.rnDeriv ν)

Proof state:
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
hνinter : ν (S ∩ T)ᶜ = 0
A : Set α
hA : MeasurableSet A
⊢ s (A ∩ (S ∩ T)) = 0

Tactic:
rw [← nonpos_iff_eq_zero]

Result:
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
hνinter : ν (S ∩ T)ᶜ = 0
A : Set α
hA : MeasurableSet A
⊢ s (A ∩ (S ∩ T)) ≤ 0

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.h
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
hνinter : ν (S ∩ T)ᶜ = 0
heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
heq' : ∀ (A : Set α), MeasurableSet A → (ν.withDensity f) A = ((ν.withDensity f).restrict (S ∩ T)) A
A : Set α
hA : MeasurableSet A
⊢ (ν.withDensity f) A = (ν.withDensity (μ.rnDeriv ν)) A

Tactic:
have hνrn : ν.withDensity (μ.rnDeriv ν) (A ∩ (S ∩ T)ᶜ) = 0 := by
  rw [← nonpos_iff_eq_zero]
  exact
    withDensity_absolutelyContinuous ν (μ.rnDeriv ν) hνinter ▸
      measure_mono inter_subset_right

Result:
case intro.intro.intro.intro.intro.intro.intro.intro.h
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
hνinter : ν (S ∩ T)ᶜ = 0
heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
heq' : ∀ (A : Set α), MeasurableSet A → (ν.withDensity f) A = ((ν.withDensity f).restrict (S ∩ T)) A
A : Set α
hA : MeasurableSet A
hνrn : (ν.withDensity (μ.rnDeriv ν)) (A ∩ (S ∩ T)ᶜ) = 0
⊢ (ν.withDensity f) A = (ν.withDensity (μ.rnDeriv ν)) A

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
⊢ ν.withDensity f = ν.withDensity (μ.rnDeriv ν)

Tactic:
rw [hadd'] at hadd

Result:
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
⊢ ν.withDensity f = ν.withDensity (μ.rnDeriv ν)

Proof state:
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
hνinter : ν (S ∩ T)ᶜ = 0
A : Set α
hA : MeasurableSet A
⊢ s (A ∩ (S ∩ T)) ≤ 0

Tactic:
exact hS₂ ▸ measure_mono (inter_subset_right.trans inter_subset_left)

Result:
no goals

Proof state:
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
⊢ ν Sᶜ + ν Tᶜ ≤ 0

Tactic:
rw [hT₃, hS₃, add_zero]

Result:
no goals

Proof state:
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hs : s ⟂ₘ ν
hadd : μ = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
⊢ ν.withDensity f = ν.withDensity (μ.rnDeriv ν)

Tactic:
obtain ⟨hmeas, hsing, hadd'⟩ := haveLebesgueDecomposition_spec μ ν

Result:
case intro.intro
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hs : s ⟂ₘ ν
hadd : μ = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hsing : μ.singularPart ν ⟂ₘ ν
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
⊢ ν.withDensity f = ν.withDensity (μ.rnDeriv ν)

Proof state:
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
hνinter : ν (S ∩ T)ᶜ = 0
A : Set α
hA : MeasurableSet A
hs : s (A ∩ (S ∩ T)) = 0
⊢ (μ.singularPart ν) (A ∩ (S ∩ T)) ≤ 0

Tactic:
exact hT₂ ▸ measure_mono (inter_subset_right.trans inter_subset_right)

Result:
no goals