Theorem: MeasureTheory.Measure.singularPart_smul
File: Mathlib/MeasureTheory/Decomposition/Lebesgue.lean

tactic: by_cases hr : r = 0
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
⊢ (r • μ).singularPart ν = r • μ.singularPart ν
state_after: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : r = 0
⊢ (r • μ).singularPart ν = r • μ.singularPart ν

case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
⊢ (r • μ).singularPart ν = r • μ.singularPart ν

tactic: by_cases hl : HaveLebesgueDecomposition μ ν
state_before: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
⊢ (r • μ).singularPart ν = r • μ.singularPart ν
state_after: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : μ.HaveLebesgueDecomposition ν
⊢ (r • μ).singularPart ν = r • μ.singularPart ν

case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : ¬μ.HaveLebesgueDecomposition ν
⊢ (r • μ).singularPart ν = r • μ.singularPart ν

tactic: rw [hr, zero_smul, zero_smul, singularPart_zero]
state_before: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : r = 0
⊢ (r • μ).singularPart ν = r • μ.singularPart ν
state_after: no goals

tactic: refine (eq_singularPart ((measurable_rnDeriv μ ν).const_smul (r : ℝ≥0∞))
      (MutuallySingular.smul r (mutuallySingular_singularPart _ _)) ?_).symm
state_before: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : μ.HaveLebesgueDecomposition ν
⊢ (r • μ).singularPart ν = r • μ.singularPart ν
state_after: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : μ.HaveLebesgueDecomposition ν
⊢ r • μ = ↑r • μ.singularPart ν + ν.withDensity (↑r • μ.rnDeriv ν)

tactic: rw [withDensity_smul _ (measurable_rnDeriv _ _), ← smul_add,
  ← haveLebesgueDecomposition_add μ ν, ENNReal.smul_def]
state_before: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : μ.HaveLebesgueDecomposition ν
⊢ r • μ = ↑r • μ.singularPart ν + ν.withDensity (↑r • μ.rnDeriv ν)
state_after: no goals

tactic: rw [singularPart, singularPart, dif_neg hl, dif_neg, smul_zero]
state_before: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : ¬μ.HaveLebesgueDecomposition ν
⊢ (r • μ).singularPart ν = r • μ.singularPart ν
state_after: case neg.hnc
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : ¬μ.HaveLebesgueDecomposition ν
⊢ ¬(r • μ).HaveLebesgueDecomposition ν

tactic: refine fun hl' ↦ hl ?_
state_before: case neg.hnc
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : ¬μ.HaveLebesgueDecomposition ν
⊢ ¬(r • μ).HaveLebesgueDecomposition ν
state_after: case neg.hnc
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : ¬μ.HaveLebesgueDecomposition ν
hl' : (r • μ).HaveLebesgueDecomposition ν
⊢ μ.HaveLebesgueDecomposition ν

tactic: rw [← inv_smul_smul₀ hr μ]
state_before: case neg.hnc
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : ¬μ.HaveLebesgueDecomposition ν
hl' : (r • μ).HaveLebesgueDecomposition ν
⊢ μ.HaveLebesgueDecomposition ν
state_after: case neg.hnc
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : ¬μ.HaveLebesgueDecomposition ν
hl' : (r • μ).HaveLebesgueDecomposition ν
⊢ (r⁻¹ • r • μ).HaveLebesgueDecomposition ν

tactic: infer_instance
state_before: case neg.hnc
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
r : ℝ≥0
hr : ¬r = 0
hl : ¬μ.HaveLebesgueDecomposition ν
hl' : (r • μ).HaveLebesgueDecomposition ν
⊢ (r⁻¹ • r • μ).HaveLebesgueDecomposition ν
state_after: no goals