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

tactic: have h_le : κ ≤ κ + η := le_add_of_nonneg_right bot_le
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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
⊢ (withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) = η
state_after: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
⊢ (withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) = η

tactic: suffices withDensity (κ + η) (fun a x ↦ Real.toNNReal (1 - rnDerivAux κ (κ + η) a x))
    + withDensity (κ + η) (fun a x ↦ Real.toNNReal (rnDerivAux κ (κ + η) a x))
    = κ + η by
  ext a s
  have h : (withDensity (κ + η) (fun a x ↦ Real.toNNReal (1 - rnDerivAux κ (κ + η) a x))
        + withDensity (κ + η) (fun a x ↦ Real.toNNReal (rnDerivAux κ (κ + η) a x))) a s
      = κ a s + η a s := by
    rw [this]
    simp
  simp only [coeFn_add, Pi.add_apply, Measure.coe_add] at h
  rwa [withDensity_rnDerivAux, add_comm, ENNReal.add_right_inj (measure_ne_top _ _)] at h
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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
⊢ (withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) = η
state_after: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
⊢ ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η

tactic: have : ∀ b, (Real.toNNReal b : ℝ≥0∞) = ENNReal.ofReal b := fun _ ↦ rfl
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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
⊢ ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η
state_after: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η

tactic: simp_rw [this, ENNReal.ofReal_sub _ (rnDerivAux_nonneg h_le), ENNReal.ofReal_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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η
state_after: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ ((withDensity (κ + η) fun a x => 1 - ENNReal.ofReal (rnDerivAux κ (κ + η) a x)) +
      withDensity (κ + η) fun a x => ENNReal.ofReal (rnDerivAux κ (κ + η) a x)) =
    κ + η

tactic: rw [withDensity_sub_add_cancel]
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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ ((withDensity (κ + η) fun a x => 1 - ENNReal.ofReal (rnDerivAux κ (κ + η) a x)) +
      withDensity (κ + η) fun a x => ENNReal.ofReal (rnDerivAux κ (κ + η) a x)) =
    κ + η
state_after: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ (withDensity (κ + η) fun a x => 1) = κ + η

case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ Measurable (Function.uncurry fun a x => 1)

case hg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ Measurable (Function.uncurry fun a x => ENNReal.ofReal (rnDerivAux κ (κ + η) a x))

case hfg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ ∀ (a : α), (fun x => ENNReal.ofReal (rnDerivAux κ (κ + η) a x)) ≤ᶠ[ae ((κ + η) a)] fun x => 1

tactic: ext a s
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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this :
  ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η
⊢ (withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) = η
state_after: case h.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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this :
  ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η
a : α
s : Set γ
a✝ : MeasurableSet s
⊢ ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) a) s = (η a) s

tactic: have h : (withDensity (κ + η) (fun a x ↦ Real.toNNReal (1 - rnDerivAux κ (κ + η) a x))
      + withDensity (κ + η) (fun a x ↦ Real.toNNReal (rnDerivAux κ (κ + η) a x))) a s
    = κ a s + η a s := by
  rw [this]
  simp
state_before: case h.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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this :
  ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η
a : α
s : Set γ
a✝ : MeasurableSet s
⊢ ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) a) s = (η a) s
state_after: case h.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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this :
  ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η
a : α
s : Set γ
a✝ : MeasurableSet s
h :
  (((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
          withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal)
        a)
      s =
    (κ a) s + (η a) s
⊢ ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) a) s = (η a) s

tactic: simp only [coeFn_add, Pi.add_apply, Measure.coe_add] at h
state_before: case h.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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this :
  ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η
a : α
s : Set γ
a✝ : MeasurableSet s
h :
  (((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
          withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal)
        a)
      s =
    (κ a) s + (η a) s
⊢ ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) a) s = (η a) s
state_after: case h.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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this :
  ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η
a : α
s : Set γ
a✝ : MeasurableSet s
h :
  ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) a) s +
      ((withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) a) s =
    (κ a) s + (η a) s
⊢ ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) a) s = (η a) s

tactic: rwa [withDensity_rnDerivAux, add_comm, ENNReal.add_right_inj (measure_ne_top _ _)] at h
state_before: case h.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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this :
  ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η
a : α
s : Set γ
a✝ : MeasurableSet s
h :
  ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) a) s +
      ((withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) a) s =
    (κ a) s + (η a) s
⊢ ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) a) s = (η a) s
state_after: no goals

tactic: rw [this]
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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this :
  ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η
a : α
s : Set γ
a✝ : MeasurableSet s
⊢ (((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
          withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal)
        a)
      s =
    (κ a) s + (η a) s
state_after: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this :
  ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η
a : α
s : Set γ
a✝ : MeasurableSet s
⊢ ((κ + η) a) s = (κ a) s + (η a) s

tactic: simp
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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this :
  ((withDensity (κ + η) fun a x => ↑(1 - rnDerivAux κ (κ + η) a x).toNNReal) +
      withDensity (κ + η) fun a x => ↑(rnDerivAux κ (κ + η) a x).toNNReal) =
    κ + η
a : α
s : Set γ
a✝ : MeasurableSet s
⊢ ((κ + η) a) s = (κ a) s + (η a) s
state_after: no goals

tactic: rw [withDensity_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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ (withDensity (κ + η) fun a x => 1) = κ + η
state_after: no goals

tactic: exact measurable_const
state_before: case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ Measurable (Function.uncurry fun a x => 1)
state_after: no goals

tactic: exact (measurable_rnDerivAux _ _).ennreal_ofReal
state_before: case hg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ Measurable (Function.uncurry fun a x => ENNReal.ofReal (rnDerivAux κ (κ + η) a x))
state_after: no goals

tactic: intro a
state_before: case hfg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ ∀ (a : α), (fun x => ENNReal.ofReal (rnDerivAux κ (κ + η) a x)) ≤ᶠ[ae ((κ + η) a)] fun x => 1
state_after: case hfg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
a : α
⊢ (fun x => ENNReal.ofReal (rnDerivAux κ (κ + η) a x)) ≤ᶠ[ae ((κ + η) a)] fun x => 1

tactic: filter_upwards [rnDerivAux_le_one h_le] with x hx
state_before: case hfg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : ↥(kernel α γ)
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
a : α
⊢ (fun x => ENNReal.ofReal (rnDerivAux κ (κ + η) a x)) ≤ᶠ[ae ((κ + η) a)] fun x => 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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
a : α
x : γ
hx : rnDerivAux κ (κ + η) a x ≤ 1 x
⊢ ENNReal.ofReal (rnDerivAux κ (κ + η) a x) ≤ 1

tactic: simp only [ENNReal.ofReal_le_one]
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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
a : α
x : γ
hx : rnDerivAux κ (κ + η) a x ≤ 1 x
⊢ ENNReal.ofReal (rnDerivAux κ (κ + η) a x) ≤ 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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
a : α
x : γ
hx : rnDerivAux κ (κ + η) a x ≤ 1 x
⊢ rnDerivAux κ (κ + η) a x ≤ 1

tactic: exact hx
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
κ η : ↥(kernel α γ)
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
a : α
x : γ
hx : rnDerivAux κ (κ + η) a x ≤ 1 x
⊢ rnDerivAux κ (κ + η) a x ≤ 1
state_after: no goals