Theorem: ProbabilityTheory.kernel.indepFun_iff_indepSet_preimage
File: Mathlib/Probability/Independence/Kernel.lean

tactic: refine indepFun_iff_measure_inter_preimage_eq_mul.trans ?_
state_before: α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsMarkovKernel κ
hf : Measurable f
hg : Measurable g
⊢ IndepFun f g κ μ ↔ ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
state_after: α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsMarkovKernel κ
hf : Measurable f
hg : Measurable g
⊢ (∀ (s : Set β) (t : Set β'),
      MeasurableSet s →
        MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)) ↔
    ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ

tactic: constructor <;> intro h s t hs ht <;> specialize h s t hs ht
state_before: α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsMarkovKernel κ
hf : Measurable f
hg : Measurable g
⊢ (∀ (s : Set β) (t : Set β'),
      MeasurableSet s →
        MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)) ↔
    ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
state_after: case mp
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsMarkovKernel κ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set β'
hs : MeasurableSet s
ht : MeasurableSet t
h : ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
⊢ IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ

case mpr
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsMarkovKernel κ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set β'
hs : MeasurableSet s
ht : MeasurableSet t
h : IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)

tactic: rwa [indepSet_iff_measure_inter_eq_mul (hf hs) (hg ht) κ μ]
state_before: case mp
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsMarkovKernel κ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set β'
hs : MeasurableSet s
ht : MeasurableSet t
h : ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
⊢ IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
state_after: no goals

tactic: rwa [← indepSet_iff_measure_inter_eq_mul (hf hs) (hg ht) κ μ]
state_before: case mpr
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsMarkovKernel κ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set β'
hs : MeasurableSet s
ht : MeasurableSet t
h : IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
state_after: no goals