-- Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean
lemma mutuallySingular_compProd_iff [SigmaFinite μ] [SigmaFinite ν] :
    μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ↔ ∀ ξ, SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η

Proof state:
case refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h : ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η
⊢ ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ ⟂ₘ ν.withDensity (μ.rnDeriv ν) ⊗ₘ η

Tactic:
refine h _ ?_ ?_ ?_

Result:
case refine_2.refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h : ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η
⊢ SFinite (ν.withDensity (μ.rnDeriv ν))

case refine_2.refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h : ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η
⊢ ν.withDensity (μ.rnDeriv ν) ≪ μ

case refine_2.refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h : ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η
⊢ ν.withDensity (μ.rnDeriv ν) ≪ ν

Proof state:
case refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h : ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ ⟂ₘ ν.withDensity (μ.rnDeriv ν) ⊗ₘ η
ξ : Measure α
hξ : SFinite ξ
hξμ : ξ ≪ μ
hξν : ξ ≪ ν
⊢ ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η

Tactic:
exact h.mono_ac ((hξμ.withDensity_rnDeriv hξν).compProd_left _)
  ((hξμ.withDensity_rnDeriv hξν).compProd_left _)

Result:
no goals

Proof state:
case refine_2.refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h : ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η
⊢ ν.withDensity (μ.rnDeriv ν) ≪ ν

Tactic:
exact withDensity_absolutelyContinuous ν (μ.rnDeriv ν)

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ ⟂ₘ ν.withDensity (μ.rnDeriv ν) ⊗ₘ η ↔
    ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η

Tactic:
refine ⟨fun h ξ hξ hξμ hξν ↦ ?_, fun h ↦ ?_⟩

Result:
case refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h : ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ ⟂ₘ ν.withDensity (μ.rnDeriv ν) ⊗ₘ η
ξ : Measure α
hξ : SFinite ξ
hξμ : ξ ≪ μ
hξν : ξ ≪ ν
⊢ ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η

case refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h : ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η
⊢ ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ ⟂ₘ ν.withDensity (μ.rnDeriv ν) ⊗ₘ η

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ↔ ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η

Tactic:
rw [(withDensity_absolutelyContinuous ν (μ.rnDeriv ν)).mutuallySingular_compProd_iff]

Result:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ ⟂ₘ ν.withDensity (μ.rnDeriv ν) ⊗ₘ η ↔
    ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)) ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ↔
    ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η

Tactic:
rw [compProd_add_left, MutuallySingular.add_left_iff]

Result:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ μ.singularPart ν ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ∧ ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ↔
    ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η

Proof state:
case refine_2.refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h : ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η
⊢ SFinite (ν.withDensity (μ.rnDeriv ν))

Tactic:
infer_instance

Result:
no goals

Proof state:
case refine_2.refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h : ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η
⊢ ν.withDensity (μ.rnDeriv ν) ≪ μ

Tactic:
exact absolutelyContinuous_of_le (withDensity_rnDeriv_le _ _)

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ μ.singularPart ν ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ∧ ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ↔
    ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η

Tactic:
simp only [(mutuallySingular_singularPart μ ν).compProd_of_left κ η, true_and]

Result:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ↔ ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η

Proof state:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ↔ ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η

Tactic:
conv_lhs => rw [μ.haveLebesgueDecomposition_add ν]

Result:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)) ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ↔
    ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η