LeanMachineLearning exposition

ProbabilityTheory.Kernel.sub_apply🔗

This page has the declaration's own card below, then its dependency graph, then a card for each dependency (type dependencies first, then the rest of the transitive closure). For a theorem, the graph and the dependency cards only follow its statement's dependencies (its proof is replaced by sorry, so what it proves doesn't depend on how); for everything else, both the type and the body/value are followed, since their content is part of what later declarations build on.

Minimal Lean file

sub_apply🔗

LemmaProbabilityTheory.Kernel.sub_apply
Details

No docstring.

🔗theorem
ProbabilityTheory.Kernel.sub_apply.{u_1, u_2} {α : Type u_1}
  {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  [MeasurableSpace.CountableOrCountablyGenerated α β] {κ η : Kernel α β}
  [(η : Kernel α β) → Decidable (IsSFiniteKernel η)] [IsFiniteKernel κ]
  [IsFiniteKernel η] (a : α) : (κ - η) a = κ a - η a
ProbabilityTheory.Kernel.sub_apply.{u_1,
    u_2}
  {α : Type u_1} {β : Type u_2}
  {mα : MeasurableSpace α}
  {mβ : MeasurableSpace β}
  [MeasurableSpace.CountableOrCountablyGenerated
      α β]
  {κ η : Kernel α β}
  [(η : Kernel α β) →
      Decidable (IsSFiniteKernel η)]
  [IsFiniteKernel κ] [IsFiniteKernel η]
  (a : α) : (κ - η) a = κ a - η a

Code

lemma sub_apply [∀ η : Kernel α β, Decidable (IsSFiniteKernel η)] [IsFiniteKernel κ]
    [IsFiniteKernel η] (a : α) :
    (κ - η) a = κ a - η a
Type uses (1)
Body uses (2)
Used by (1)

Actions: Source · Open Issue

Proof
by
  ext s hs
  rw [sub_apply_eq_rnDeriv_add_singularPart, Kernel.withDensity_apply _ (by fun_prop),
    Kernel.singularPart_eq_singularPart_measure, Measure.sub_apply_eq_rnDeriv_add_singularPart _]
  simp only [Measure.coe_add, Pi.add_apply]
  congr 2
  refine MeasureTheory.withDensity_congr_ae ?_
  filter_upwards [Kernel.rnDeriv_eq_rnDeriv_measure (κ := κ) (η := η) (a := a)] with b hb
  simp [← hb]

Dependency graph

Type dependencies (1)

instSubOfDecidableIsSFiniteKernel_leanMachineLearning🔗

InstanceProbabilityTheory.Kernel.instSubOfDecidableIsSFiniteKernel_leanMachineLearning
Details

No docstring.

🔗def
ProbabilityTheory.Kernel.instSubOfDecidableIsSFiniteKernel_leanMachineLearning.{u_1,
    u_2}
  {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α}
  {mβ : MeasurableSpace β}
  [MeasurableSpace.CountableOrCountablyGenerated α β]
  [(η : Kernel α β) → Decidable (IsSFiniteKernel η)] : Sub (Kernel α β)
ProbabilityTheory.Kernel.instSubOfDecidableIsSFiniteKernel_leanMachineLearning.{u_1,
    u_2}
  {α : Type u_1} {β : Type u_2}
  {mα : MeasurableSpace α}
  {mβ : MeasurableSpace β}
  [MeasurableSpace.CountableOrCountablyGenerated
      α β]
  [(η : Kernel α β) →
      Decidable (IsSFiniteKernel η)] :
  Sub (Kernel α β)

Code

noncomputable
instance [∀ η : Kernel α β, Decidable (IsSFiniteKernel η)] :
    Sub (Kernel α β) where
  sub κ η
Used by (11)

Actions: Source · Open Issue

Proof
if h : IsSFiniteKernel κ ∧ IsSFiniteKernel η
    then
      have := h.1
      have := h.2
      η.withDensity (fun a ↦ κ.rnDeriv η a - 1) + κ.singularPart η
    else 0