Theorem: ProbabilityTheory.kernel.rnDerivAux_le_one
File: Mathlib/Probability/Kernel/RadonNikodym.lean

tactic: filter_upwards [Measure.rnDeriv_le_one_of_le (hκη a)] with x hx_le_one
state_before: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
⊢ rnDerivAux κ η a ≤ᶠ[ae (η a)] 1
state_after: case h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
⊢ rnDerivAux κ η a x ≤ 1 x

tactic: simp_rw [rnDerivAux]
state_before: case h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
⊢ rnDerivAux κ η a x ≤ 1 x
state_after: case h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
⊢ (if hα : Countable α then ((∂κ a/∂η a) x).toReal else density (map κ (fun a => (a, ())) ⋯) η a x univ) ≤ 1 x

tactic: split_ifs with hα
state_before: case h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
⊢ (if hα : Countable α then ((∂κ a/∂η a) x).toReal else density (map κ (fun a => (a, ())) ⋯) η a x univ) ≤ 1 x
state_after: case pos
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
hα : Countable α
⊢ ((∂κ a/∂η a) x).toReal ≤ 1 x

case neg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
hα : ¬Countable α
⊢ density (map κ (fun a => (a, ())) ⋯) η a x univ ≤ 1 x

tactic: refine ENNReal.toReal_le_of_le_ofReal zero_le_one ?_
state_before: case pos
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
hα : Countable α
⊢ ((∂κ a/∂η a) x).toReal ≤ 1 x
state_after: case pos
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
hα : Countable α
⊢ (∂κ a/∂η a) x ≤ ENNReal.ofReal (1 x)

tactic: simp only [Pi.one_apply, ENNReal.ofReal_one]
state_before: case pos
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
hα : Countable α
⊢ (∂κ a/∂η a) x ≤ ENNReal.ofReal (1 x)
state_after: case pos
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
hα : Countable α
⊢ (∂κ a/∂η a) x ≤ 1

tactic: exact hx_le_one
state_before: case pos
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
hα : Countable α
⊢ (∂κ a/∂η a) x ≤ 1
state_after: no goals

tactic: have := hαγ.countableOrCountablyGenerated.resolve_left hα
state_before: case neg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
hα : ¬Countable α
⊢ density (map κ (fun a => (a, ())) ⋯) η a x univ ≤ 1 x
state_after: case neg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
hα : ¬Countable α
this : MeasurableSpace.CountablyGenerated γ
⊢ density (map κ (fun a => (a, ())) ⋯) η a x univ ≤ 1 x

tactic: exact density_le_one ((fst_map_id_prod _ measurable_const).trans_le hκη) _ _ _
state_before: case neg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝ : IsFiniteKernel η
hκη : κ ≤ η
a : α
x : γ
hx_le_one : (∂↑κ a/∂η a) x ≤ 1 x
hα : ¬Countable α
this : MeasurableSpace.CountablyGenerated γ
⊢ density (map κ (fun a => (a, ())) ⋯) η a x univ ≤ 1 x
state_after: no goals