actual_bounded_path_functional_expansion

Plain-English statement

- Full bounded signed path expansion for the actual stopped-clock process.

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_bounded_path_functional_expansion
    {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_bounded_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))
  let wgt : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) → Prefix n T → ℝ :=
    fun y xRef z₀ n T t =>
      Real.exp (-(∫ s in (0 : ℝ)..(T : ℝ),
        rate xRef (Γ y xRef z₀ n T t ⟨min (Real.toNNReal s) T, min_le_right _ _⟩))) *
      ∏ 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))
  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) ∧
      (∀ y xRef : E, ∀ z₀ : E × E, ∀ T : ℝ≥0,
        let ZT : (ℕ → ℝ) → Path T := fun u s => Z y xRef z₀ s.val u
        ∀ F : Path T → ℝ, Measurable F → ∀ M : ℝ, 0 ≤ M →
          (∀ p : Path T, |F p| ≤ M) →
          Integrable (fun u => F (ZT u)) P ∧
          (∀ n : ℕ, Integrable (fun t : Prefix n T =>
            F (Γ y xRef z₀ n T t) * wgt y xRef z₀ n T t) (m n T)) ∧
          Summable (fun n : ℕ => |∫ t : Prefix n T,
            F (Γ y xRef z₀ n T t) * wgt y xRef z₀ n T t ∂m n T|) ∧
          (∫ u, F (ZT u) ∂P) = ∑' n : ℕ, ∫ t : Prefix n T,
            F (Γ y xRef z₀ n T t) * wgt y xRef z₀ n T t ∂m n T)

  obtain ⟨Z, hZM, hZarc, hZfallback, hZzero, hAll⟩ :=
    ActualFiniteCountPath.actual_finite_count_path_law hα hαβ hV hH hη hβη
  refine ⟨Z, hZM, hZarc, hZfallback, hZzero, ?_⟩
  intro y xRef z₀ T
  let ZT : (ℕ → ℝ) → Path T := fun u s => Z y xRef z₀ s.val u
  dsimp only
  intro F hF M hM hbound
  letI : IsProbabilityMeasure P :=
    AutoSamplingTheory.TechnicalLemmas.Probability.UnitExponentialProduct.unit_exponential_product_laws.1
  have hbase := (hAll 0 T).2.2.2 y xRef z₀
  have hZT : Measurable ZT := hbase.2.1
  have hN : Measurable (N y xRef z₀ T) := hbase.2.2.1
  have hi : Integrable (fun u => F (ZT u)) P :=
    ⟨(hF.comp hZT).aestronglyMeasurable,
      HasFiniteIntegral.of_bounded (Filter.Eventually.of_forall
        (fun u => by simpa only [Real.norm_eq_abs] using hbound (ZT u)))⟩
  have hsector (n : ℕ) :
      Integrable (fun t : Prefix n T => F (Γ y xRef z₀ n T t) *
        wgt y xRef z₀ n T t) (m n T) ∧
      (∫ u in {u | N y xRef z₀ T u = n}, F (ZT u) ∂P) =
        ∫ t : Prefix n T, F (Γ y xRef z₀ n T t) * wgt y xRef z₀ n T t ∂m n T := by
    have hn := hAll n T
    obtain ⟨hHM, hZTM, hNM, hCM, hinterp, hident, hAE, hdisj, hcover,
      hlaw, htests, hmass, hmassbound⟩ := hn.2.2.2 y xRef z₀
    have hΓ : Measurable (Γ y xRef z₀ n T) := hn.2.1.comp
      (f := fun t : Prefix n T => (((y,xRef),z₀),t))
      (measurable_const.prodMk measurable_id)
    have hρ : Measurable (ρ y xRef z₀ n T) := hn.2.2.1.comp
      (f := fun t : Prefix n T => (((y,xRef),z₀),t))
      (measurable_const.prodMk measurable_id)
    have hK : Measurable (K y xRef z₀ n T) :=
      (ActualOrderedPrefix.actual_ordered_prefix_law hα hαβ hV hH hη hβη n T).2.2.1.comp
        (f := fun t : Prefix n T => (((y,xRef),z₀),t))
        (measurable_const.prodMk measurable_id)
    have hw (t : Prefix n T) : 0 ≤ wgt y xRef z₀ n T t := by
      apply mul_nonneg (Real.exp_nonneg _)
      exact Finset.prod_nonneg (fun i _ => mul_nonneg (Real.sqrt_nonneg _) (le_max_left _ _))
    have heq (t : Prefix n T) :
        ρ y xRef z₀ n T t = ENNReal.ofReal (wgt y xRef z₀ n T t) := by
      have hd := (ActualPrescribedPathDensity.actual_prescribed_path_density_rewrite
        hα hαβ hV hH hη hβη y xRef z₀ n T t).2.2.2
      simpa only [Fin.snoc_castSucc] using hd
    exact signed_sector hHM hZTM hK hΓ hρ hw heq hlaw hF hi.integrableOn
  have hsum := signed_count_partition hN hi
  have heq (n : ℕ) := (hsector n).2
  refine ⟨hi, fun n => (hsector n).1, ?_, ?_⟩
  · simpa only [heq] using hsum.1
  · simpa only [heq] using hsum.2

#print axioms actual_bounded_path_functional_expansion

end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualBoundedPathExpansion

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualBoundedPathExpansion.lean:167published source at ed709a7f320d

Proof architecture

Actual PBPS bounded path expectations; 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.
`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.
`simpa` closes the goal after a controlled simplification of a typed result.

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.