awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue (goal)
Goal:
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 : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
A : Set α
hA : MeasurableSet A
hs : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞)
hsing : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞)
h :
  Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    (((ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
h₁ :
  Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₂ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞) =
    (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
h₃ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
      ((s : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) =
    (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
h₄ :
  Eq (α := ℝ≥0∞) ((ν.withDensity f + s : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₅ :
  Eq (α := ℝ≥0∞) ((s + ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) (((ν.withDensity f).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    (((ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
have h₆ : Eq (α := ℝ≥0∞) ((μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
  ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
Tactic: rw [restrict_apply hA, restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hs, ←
      add_apply, add_comm, ← hadd, add_apply, hsing, zero_add]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue (goal)
Goal:
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 : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
A : Set α
hA : MeasurableSet A
hs : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞)
hsing : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞)
h :
  Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    (((ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
h₁ :
  Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₂ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞) =
    (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
h₃ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
      ((s : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) =
    (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
h₄ :
  Eq (α := ℝ≥0∞) ((ν.withDensity f + s : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₅ :
  Eq (α := ℝ≥0∞) ((s + ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₆ :
  Eq (α := ℝ≥0∞) ((μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) (((ν.withDensity f).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    (((ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
have h₇ : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) =
  (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
Tactic: rw [restrict_apply hA, restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hs, ←
      add_apply, add_comm, ← hadd, add_apply, hsing, zero_add]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue (goal)
Goal:
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 : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
A : Set α
hA : MeasurableSet A
hs : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞)
hsing : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞)
h :
  Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    (((ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
h₁ :
  Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₂ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞) =
    (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
h₃ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
      ((s : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) =
    (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
h₄ :
  Eq (α := ℝ≥0∞) ((ν.withDensity f + s : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₅ :
  Eq (α := ℝ≥0∞) ((s + ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₆ :
  Eq (α := ℝ≥0∞) ((μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₇ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
      ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) =
    (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
⊢ Eq (α := ℝ≥0∞) (((ν.withDensity f).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    (((ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
have h₈ : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) (0 : ℝ≥0∞) ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) =
  (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
Tactic: rw [restrict_apply hA, restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hs, ←
      add_apply, add_comm, ← hadd, add_apply, hsing, zero_add]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue (hypothesis)
Goal:
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 : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ ν.withDensity f = ν.withDensity (μ.rnDeriv ν)
have heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
Tactic: have heq :
    (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) := by
    ext1 A hA
    have hs : s (A ∩ (S ∩ T)) = 0 := by
      rw [← nonpos_iff_eq_zero]
      exact hS₂ ▸ measure_mono (inter_subset_right.trans inter_subset_left)
    have hsing : μ.singularPart ν (A ∩ (S ∩ T)) = 0 := by
      rw [← nonpos_iff_eq_zero]
      exact hT₂ ▸ measure_mono (inter_subset_right.trans inter_subset_right)
    rw [restrict_apply hA, restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hs, ←
      add_apply, add_comm, ← hadd, add_apply, hsing, zero_add]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue (goal)
Goal:
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 : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
A : Set α
hA : MeasurableSet A
⊢ Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)
have rw : LE.le (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: rw [← nonpos_iff_eq_zero]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue (hypothesis)
Goal:
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 : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
A : Set α
hA : MeasurableSet A
⊢ Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    (((ν.withDensity f).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
have hνfinter : Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: have hνfinter : ν.withDensity f (A ∩ (S ∩ T)ᶜ) = 0 := by
      rw [← nonpos_iff_eq_zero]
      exact withDensity_absolutelyContinuous ν f hνinter ▸ measure_mono inter_subset_right
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue (goal)
Goal:
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 : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
A : Set α
hA : MeasurableSet A
hνfinter : Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    (((ν.withDensity f).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
have rw : Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
  ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
Tactic: rw [restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hνfinter, ← diff_eq,
      measure_inter_add_diff _ (hS₁.inter hT₁)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue (goal)
Goal:
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 : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
A : Set α
hA : MeasurableSet A
hνfinter : Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)
rw :
  Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    (((ν.withDensity f).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
have rw₁ : (ν.withDensity f : (a : Set α) → ℝ≥0∞) A =
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: rw [restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hνfinter, ← diff_eq,
      measure_inter_add_diff _ (hS₁.inter hT₁)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue (goal)
Goal:
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 : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
A : Set α
hA : MeasurableSet A
hνfinter : Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)
rw :
  Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
rw₁ :
  (ν.withDensity f : (a : Set α) → ℝ≥0∞) A =
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    (((ν.withDensity f).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
have rw₂ : (ν.withDensity f : (a : Set α) → ℝ≥0∞) A =
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)ᶜ) : ℝ≥0∞)
Tactic: rw [restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hνfinter, ← diff_eq,
      measure_inter_add_diff _ (hS₁.inter hT₁)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue (goal)
Goal:
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 : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
A : Set α
hA : MeasurableSet A
hνfinter : Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)
rw :
  Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
rw₁ :
  (ν.withDensity f : (a : Set α) → ℝ≥0∞) A =
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞)
rw₂ :
  (ν.withDensity f : (a : Set α) → ℝ≥0∞) A =
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
      ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)ᶜ) : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    (((ν.withDensity f).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
have rw₃ : (ν.withDensity f : (a : Set α) → ℝ≥0∞) A =
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A \ (S ∩ T)) : ℝ≥0∞)
Tactic: rw [restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hνfinter, ← diff_eq,
      measure_inter_add_diff _ (hS₁.inter hT₁)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue (goal)
Goal:
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 : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
A : Set α
hA : MeasurableSet A
hνfinter : Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)ᶜ) : ℝ≥0∞) (0 : ℝ≥0∞)
rw :
  Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
rw₁ :
  (ν.withDensity f : (a : Set α) → ℝ≥0∞) A =
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞)
rw₂ :
  (ν.withDensity f : (a : Set α) → ℝ≥0∞) A =
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
      ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)ᶜ) : ℝ≥0∞)
rw₃ :
  (ν.withDensity f : (a : Set α) → ℝ≥0∞) A =
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
      ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A \ (S ∩ T)) : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    (((ν.withDensity f).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
have rw₄ : Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
Tactic: rw [restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hνfinter, ← diff_eq,
      measure_inter_add_diff _ (hS₁.inter hT₁)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue (hypothesis)
Goal:
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 : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T)
⊢ ν.withDensity f = ν.withDensity (μ.rnDeriv ν)
have heq' : ∀ (A : Set α),
  MeasurableSet A →
    Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
      (((ν.withDensity f).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
Tactic: have heq' :
    ∀ A : Set α, MeasurableSet A → ν.withDensity f A = (ν.withDensity f).restrict (S ∩ T) A := by
    intro A hA
    have hνfinter : ν.withDensity f (A ∩ (S ∩ T)ᶜ) = 0 := by
      rw [← nonpos_iff_eq_zero]
      exact withDensity_absolutelyContinuous ν f hνinter ▸ measure_mono inter_subset_right
    rw [restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hνfinter, ← diff_eq,
      measure_inter_add_diff _ (hS₁.inter hT₁)]