actual_finite_count_path_law

Plain-English statement

- Exact sealed finite-count law; every probability/process ingredient is derived in the proof.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem actual_finite_count_path_law
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : ℝ≥0} {η : ℝ}
    (hα : 0 < (α : ℝ)) (hαβ : α ≤ β) (hV : ContDiff ℝ 2 V)
    (hH : ∀ x v : E,
      (α : ℝ) * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ V) x v) v ∧
      (fderiv ℝ (fderiv ℝ V) x v) v ≤ (β : ℝ) * ‖v‖ ^ 2)
    (hη : 0 < η) (hβη : (β : ℝ) * η ≤ 1) :
    actual_finite_count_path_statement hα hαβ hV hH hη hβη := by
  classical
  let c : E → E → E := fun y xRef => y - η • gradient V xRef
  let Φ : E → E → ℝ → (E × E) → (E × E) := fun y xRef t z =>
    (c y xRef + Real.cos t • (z.1 - c y xRef) +
        (Real.sqrt η * Real.sin t) • z.2,
     (-Real.sin t / Real.sqrt η) • (z.1 - c y xRef) + Real.cos t • z.2)
  let S : E → (E × E) → (E × E) := fun xRef z =>
    (z.1, z.2 - (2 * inner ℝ z.2 (gradient V z.1 - gradient V xRef) /
      ‖gradient V z.1 - gradient V xRef‖ ^ 2) • (gradient V z.1 - gradient V xRef))
  let rate : E → (E × E) → ℝ := fun xRef z =>
    Real.sqrt η * max 0 (inner ℝ z.2 (gradient V z.1 - gradient V xRef))
  let Λ : E → E → (E × E) → ℝ≥0 → ℝ := fun y xRef z t =>
    ∫ s in (0 : ℝ)..(t : ℝ), rate xRef (Φ y xRef s z)
  let τ : E → E → (E × E) → ℝ≥0 → WithTop ℝ≥0 := fun y xRef z e =>
    MeasureTheory.hittingAfter
      (fun t : ℝ≥0 => fun a : (E × E) × (E × E) × ℝ≥0 =>
        Λ a.1.1 a.1.2 a.2.1 t - (a.2.2 : ℝ))
      (Set.Ici (0 : ℝ)) 0 ((y, xRef), z, e)
  let next : E → E → ℝ≥0 → ((ℝ≥0 × (E × E)) ⊕ Unit) →
      ((ℝ≥0 × (E × E)) ⊕ Unit) := fun y xRef e r =>
    match r with
    | Sum.inr _ => Sum.inr ()
    | Sum.inl a =>
      if τ y xRef a.2 e = ⊤ then Sum.inr () else
        Sum.inl (a.1 + (τ y xRef a.2 e).untopD 0,
          S xRef (Φ y xRef (((τ y xRef a.2 e).untopD 0 : ℝ≥0) : ℝ) a.2))
  let record : E → E → (E × E) → (ℕ → ℝ≥0) → ℕ →
      ((ℝ≥0 × (E × E)) ⊕ Unit) := fun y xRef z₀ e n =>
    Nat.rec (Sum.inl (0, z₀)) (fun k r => next y xRef (e k) r) n
  let P : Measure (ℕ → ℝ) := Measure.infinitePi (fun _ : ℕ => expMeasure (1 : ℝ))
  let ε : (ℕ → ℝ) → ℕ → ℝ≥0 := fun w k => Real.toNNReal (w k)
  let history : E → E → (E × E) → (n : ℕ) → (ℕ → ℝ) →
      (Fin (n + 1) → ((ℝ≥0 × (E × E)) ⊕ Unit)) :=
    fun y xRef z₀ n w j => record y xRef z₀ (ε w) j.val
  let Rec := (ℝ≥0 × (E × E)) ⊕ Unit
  let Hist : ℕ → Type _ := fun n => Fin (n + 1) → Rec
  let readTime : Rec → ℝ := fun r => match r with
    | Sum.inl q => (q.1 : ℝ)
    | Sum.inr _ => 0
  let Δ : (n : ℕ) → ℝ≥0 → Set (Fin n → ℝ) := fun n T =>
    {t | StrictMono (Fin.cons (0 : ℝ) t) ∧ ∀ i, t i < (T : ℝ)}
  let Prefix : (n : ℕ) → ℝ≥0 → Type := fun n T => {t : Fin n → ℝ // t ∈ Δ n T}
  let m : (n : ℕ) → (T : ℝ≥0) → Measure (Prefix n T) := fun n T =>
    Measure.comap (fun t : Prefix n T => t.val) (volume : Measure (Fin n → ℝ))
  let zAt : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) →
      Prefix n T → Fin (n + 1) → (E × E) := fun y xRef z₀ n T t =>
    Fin.induction z₀ (fun i z =>
      S xRef (Φ y xRef (t.val i - (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) i.castSucc) z))
  let K : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) →
      Prefix n T → Hist n := fun y xRef z₀ n T t j =>
    Sum.inl (Real.toNNReal ((Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) j), zAt y xRef z₀ n T t j)
  let qReal : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) →
      Prefix n T → ℝ := fun y xRef z₀ n T t =>
    ∏ i : Fin n,
      rate xRef (Φ y xRef (t.val i - (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) i.castSucc)
        (zAt y xRef z₀ n T t i.castSucc)) *
      Real.exp (-Λ y xRef (zAt y xRef z₀ n T t i.castSucc)
        (Real.toNNReal (t.val i - (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) i.castSucc)))
  let q : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) →
      Prefix n T → ℝ≥0∞ := fun y xRef z₀ n T t =>
    ENNReal.ofReal (qReal y xRef z₀ n T t)
  let A : (n : ℕ) → ℝ≥0 → Set (Hist n) := fun n T =>
    {h | (∀ j, ∃ a : ℝ≥0 × (E × E), h j = Sum.inl a) ∧
      StrictMono (fun j => readTime (h j)) ∧
      ∀ i : Fin n, readTime (h i.succ) < (T : ℝ)}
  let eventTime : E → E → (E × E) → (ℕ → ℝ) → ℕ → WithTop ℝ≥0 :=
    fun y xRef z₀ w j => (record y xRef z₀ (ε w) j).elim
      (fun a => (a.1 : WithTop ℝ≥0)) (fun _ => ⊤)
  let I : E → E → (E × E) → ℝ≥0 → ℕ → Set (ℕ → ℝ) :=
    fun y xRef z₀ T n => {w | eventTime y xRef z₀ w n ≤ (T : WithTop ℝ≥0) ∧
      (T : WithTop ℝ≥0) < eventTime y xRef z₀ w (n + 1)}
  let N : E → E → (E × E) → ℝ≥0 → (ℕ → ℝ) → ℕ := fun y xRef z₀ T w =>
    if h : ∃ n, w ∈ I y xRef z₀ T n then Nat.find h else 0
  let liveTime : Rec → ℝ≥0 := fun r => r.elim Prod.fst (fun _ => 0)
  let d : (n : ℕ) → ℝ≥0 → Hist n → ℝ≥0 :=
    fun n T h => T - liveTime (h ⟨n, Nat.lt_succ_self n⟩)
  let survival : E → E → Rec → ℝ≥0 → ℝ → Prop :=
    fun y xRef r u e => r.elim
      (fun a => (u : WithTop ℝ≥0) < τ y xRef a.2 (Real.toNNReal e)) (fun _ => False)
  let weight : E → E → Rec → ℝ≥0 → ℝ≥0∞ :=
    fun y xRef r u => r.elim
      (fun a => ENNReal.ofReal (Real.exp (-Λ y xRef a.2 u))) (fun _ => 0)
  let C : E → E → (E × E) → (n : ℕ) → ℝ≥0 → Set (ℕ → ℝ) :=
    fun y xRef z₀ n T => {w | history y xRef z₀ n w ∈ A n T ∧
      survival y xRef (record y xRef z₀ (ε w) n) (d n T (history y xRef z₀ n w)) (w n)}
  let Path : ℝ≥0 → Type _ := fun T => {s : ℝ≥0 // s ≤ T} → E × E
  let Ψ : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) → Hist n → Path T :=
    fun y xRef z₀ n T h s =>
      Fin.foldl (n + 1) (fun z j => (h j).elim
        (fun a => if a.1 ≤ s.val then Φ y xRef ((s.val - a.1 : ℝ≥0) : ℝ) a.2 else z)
        (fun _ => z)) z₀
  let Γ : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) → Prefix n T → Path T :=
    fun y xRef z₀ n T t => Ψ y xRef z₀ n T (K y xRef z₀ n T t)
  let ρ : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) → Prefix n T → ℝ≥0∞ :=
    fun y xRef z₀ n T t => q y xRef z₀ n T t *
      weight y xRef ((K y xRef z₀ n T t) ⟨n, Nat.lt_succ_self n⟩)
        (d n T (K y xRef z₀ n T t))
  change
    ∃ Z : E → E → (E × E) → ℝ≥0 → (ℕ → ℝ) → (E × E),
      Measurable (fun a : ((E × E) × (E × E)) × ℝ≥0 × (ℕ → ℝ) =>
        Z a.1.1.1 a.1.1.2 a.1.2 a.2.1 a.2.2) ∧
      (∀ y xRef : E, ∀ z₀ : E × E, ∀ w : ℕ → ℝ, ∀ s : ℝ≥0,
        ∀ n : ℕ, ∀ a : ℝ≥0 × (E × E), w ∈ I y xRef z₀ s n →
          record y xRef z₀ (ε w) n = Sum.inl a →
          Z y xRef z₀ s w = Φ y xRef ((s - a.1 : ℝ≥0) : ℝ) a.2) ∧
      (∀ y xRef : E, ∀ z₀ : E × E, ∀ w : ℕ → ℝ, ∀ s : ℝ≥0,
        (¬ ∃ n, w ∈ I y xRef z₀ s n) → Z y xRef z₀ s w = z₀) ∧
      (∀ y xRef : E, ∀ z₀ : E × E, ∀ᵐ w ∂P, Z y xRef z₀ 0 w = z₀) ∧
      (∀ n : ℕ, ∀ T : ℝ≥0,
        Measurable (fun p : ((E × E) × (E × E)) × Hist n =>
          Ψ p.1.1.1 p.1.1.2 p.1.2 n T p.2) ∧
        Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
          Γ p.1.1.1 p.1.1.2 p.1.2 n T p.2) ∧
        Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
          ρ p.1.1.1 p.1.1.2 p.1.2 n T p.2) ∧
        (∀ y xRef : E, ∀ z₀ : E × E,
          let H := history y xRef z₀ n
          let ZT : (ℕ → ℝ) → Path T := fun w s => Z y xRef z₀ s.val w
          Measurable H ∧ Measurable ZT ∧ Measurable (N y xRef z₀ T) ∧
          MeasurableSet (C y xRef z₀ n T) ∧
          (∀ t : Prefix n T, ∀ s : {s : ℝ≥0 // s ≤ T}, ∀ j : Fin (n + 1),
            (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) j ≤ (s.val : ℝ) →
            (∀ k : Fin (n + 1), j < k → (s.val : ℝ) < (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) k) →
            Γ y xRef z₀ n T t s =
              Φ y xRef ((s.val : ℝ) - (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) j) (zAt y xRef z₀ n T t j)) ∧
          (∀ w ∈ C y xRef z₀ n T, Ψ y xRef z₀ n T (H w) = ZT w) ∧
          (∀ᵐ w ∂P, (∀ j : ℕ, 0 < j → eventTime y xRef z₀ w j ≠ (T : WithTop ℝ≥0)) ∧
            (∀ k : ℕ, N y xRef z₀ T w = k ↔ w ∈ C y xRef z₀ k T) ∧
            {j : ℕ | 0 < j ∧ eventTime y xRef z₀ w j < (T : WithTop ℝ≥0)}.Finite ∧
            {j : ℕ | 0 < j ∧ eventTime y xRef z₀ w j ≤ (T : WithTop ℝ≥0)}.Finite ∧
            N y xRef z₀ T w = {j : ℕ | 0 < j ∧ eventTime y xRef z₀ w j < (T : WithTop ℝ≥0)}.ncard ∧
            N y xRef z₀ T w = {j : ℕ | 0 < j ∧ eventTime y xRef z₀ w j ≤ (T : WithTop ℝ≥0)}.ncard) ∧
          (Pairwise fun i j : ℕ => Disjoint (C y xRef z₀ i T) (C y xRef z₀ j T)) ∧
          P (⋃ k : ℕ, C y xRef z₀ k T) = 1 ∧
          Measure.map (fun w => (H w, ZT w)) (P.restrict {w | N y xRef z₀ T w = n}) =
            Measure.map (fun t => (K y xRef z₀ n T t, Γ y xRef z₀ n T t))
              ((m n T).withDensity (ρ y xRef z₀ n T)) ∧
          (∀ F : Hist n × Path T → ℝ≥0∞, Measurable F →
            (∫⁻ w in {w | N y xRef z₀ T w = n}, F (H w, ZT w) ∂P) =
              ∫⁻ t : Prefix n T, F (K y xRef z₀ n T t, Γ y xRef z₀ n T t) *
                ρ y xRef z₀ n T t ∂m n T) ∧
          (∫⁻ t : Prefix n T, ρ y xRef z₀ n T t ∂m n T) = P {w | N y xRef z₀ T w = n} ∧
          P {w | N y xRef z₀ T w = n} ≤ 1))
  obtain ⟨Z, hZM, hZarc, hZfallback, hZzero, hZpath⟩ :=
    actual_history_path_identification hα hαβ hV hH hη hβη
  refine ⟨Z, hZM, hZarc, hZfallback, hZzero, ?_⟩
  intro n T
  have hΨ := actual_history_interpolation_measurable hα hαβ hV hH hη hβη n T
  have hW := actual_terminal_weighted_prefix_tests hα hαβ hV hH hη hβη n T
  refine ⟨hΨ, hW.1, hW.2.1, ?_⟩
  intro y xRef z₀
  let H := history y xRef z₀ n
  let ZT : (ℕ → ℝ) → Path T := fun w s => Z y xRef z₀ s.val w
  have hWL := hW.2.2 y xRef z₀
  have hHM : Measurable H :=
    ((AutoSamplingTheory.ExampleCases.ProximalBPS.ActualOrderedPrefix.actual_ordered_prefix_law
      hα hαβ hV hH hη hβη n T).2.2.2.2 y xRef z₀).1
  have hZT : Measurable ZT := by
    apply measurable_pi_lambda
    intro s
    exact hZM.comp (f := fun w : ℕ → ℝ => (((y,xRef),z₀),(s.val,w)))
      (measurable_const.prodMk (measurable_const.prodMk measurable_id))
  have hNM : Measurable (N y xRef z₀ T) :=
    (actual_count_selector_measurable hα hαβ hV hH hη hβη).comp
      (f := fun w : ℕ → ℝ => (((y,xRef),z₀),(T,w)))
      (measurable_const.prodMk (measurable_const.prodMk measurable_id))
  have hc := actual_count_event_identification hα hαβ hV hH hη hβη y xRef z₀ T
  have heq : {w | N y xRef z₀ T w = n} =ᵐ[P] C y xRef z₀ n T := by
    filter_upwards [hc.2.1] with w hw
    exact propext (hw.2.1 n)
  have hr : P.restrict {w | N y xRef z₀ T w = n} = P.restrict (C y xRef z₀ n T) :=
    Measure.restrict_congr_set heq
  have hp : ∀ w ∈ C y xRef z₀ n T, Ψ y xRef z₀ n T (H w) = ZT w :=
    hZpath n T y xRef z₀
  refine ⟨hHM, hZT, hNM, hWL.1,
    actual_prescribed_path_interpolation hα hαβ hV hH hη hβη n T y xRef z₀,
    hp, hc.2.1, hc.2.2.1, hc.2.2.2, ?_, ?_, ?_, ?_⟩
  · rw [hr]
    calc
      Measure.map (fun w => (H w, ZT w)) (P.restrict (C y xRef z₀ n T)) =
          Measure.map (fun w => (H w, Ψ y xRef z₀ n T (H w))) (P.restrict (C y xRef z₀ n T)) := by
        apply Measure.map_congr
        exact ae_restrict_of_forall_mem hWL.1 (fun w hw => Prod.ext rfl (hp w hw).symm)
      _ = _ := hWL.2.2.2.1
  · intro F hF
    rw [hr]
    calc
      (∫⁻ w, F (H w, ZT w) ∂P.restrict (C y xRef z₀ n T)) =
          ∫⁻ w, F (H w, Ψ y xRef z₀ n T (H w)) ∂P.restrict (C y xRef z₀ n T) := by
        apply lintegral_congr_ae
        exact ae_restrict_of_forall_mem hWL.1 (fun w hw => congrArg (fun z => F (H w,z)) (hp w hw).symm)
      _ = _ := hWL.2.2.1 F hF
  · exact hWL.2.2.2.2.1.trans (measure_congr heq).symm
  · rw [measure_congr heq]
    exact hWL.2.2.2.2.2

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteCountPath

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualFiniteCountPath.lean:2005published source at ed709a7f320d

Proof architecture

Actual PBPS finite path-law substrate; no SALD claim

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`calc` records an equality or inequality chain matching a paper calculation.
`rw` rewrites by an established identity.
`apply` reduces the goal to the hypotheses of a reusable theorem.
`refine` instantiates a reusable theorem while leaving explicit subgoals.
`exact` closes the current goal with an already typed term.

Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.
Totalized `fderiv` values must not be read as a differentiability theorem.
Almost-everywhere hypotheses depend on the stated measure and representative.
Density statements retain normalization and absolute-continuity prerequisites.