actual_countable_endpoint_law

Plain-English statement

- The actual Gibbs-initialized endpoint law is the countable sector sum, a probability, and preserved by the flipped swap.

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_countable_endpoint_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_countable_endpoint_statement hα hαβ hV hH hη hβη := by
  classical
  unfold actual_countable_endpoint_statement
  intro c Φ S rate Λ τ next record P ε history Rec Hist readTime Δ Prefix m zAt K qReal q A eventTime I N liveTime d survival weight C Path Ψ Γ ρ R Vy nu
  obtain ⟨Z, hZm, hZa, hZf, hZ0, h93⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteCountPath.actual_finite_count_path_law
      hα hαβ hV hH hη hβη
  refine ⟨Z, hZm, hZa, hZf, hZ0, ?_⟩
  intro y r T B sigma L M Q J
  have h103 := AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseNormalization.actual_phase_gibbs_normalization
    hα hαβ hV hH hη hβη y
  letI : IsProbabilityMeasure (nu y) := by
    have hp := h103.2.2.2.2.2.1
    rw [h103.2.2.2.2.2.2] at hp
    exact hp
  letI : IsProbabilityMeasure P :=
    AutoSamplingTheory.TechnicalLemmas.Probability.UnitExponentialProduct.unit_exponential_product_laws.1
  have hembZ : Measurable (fun q : (E × E) × (ℕ → ℝ) => (((y, r), q.1), T, q.2)) :=
    (measurable_const.prodMk measurable_fst).prodMk (measurable_const.prodMk measurable_snd)
  have hZend := hZm.comp hembZ
  have hM : Measurable M := measurable_fst.prodMk hZend
  have hQ : IsProbabilityMeasure Q := Measure.isProbabilityMeasure_map hM.aemeasurable
  have hB (n : ℕ) : Measurable (B n) := by
    have hemb : Measurable (fun q : (E × E) × Prefix n T => (((y, r), q.1), q.2)) :=
      (measurable_const.prodMk measurable_fst).prodMk measurable_snd
    have hp := (h93 n T).2.1.comp hemb
    have he := (measurable_pi_apply (⟨T, le_rfl⟩ : {s : ℝ≥0 // s ≤ T})).comp hp
    exact measurable_fst.prodMk he
  have hρ (n : ℕ) : Measurable (fun q : (E × E) × Prefix n T => ρ y r q.1 n T q.2) := by
    have hemb : Measurable (fun q : (E × E) × Prefix n T => (((y, r), q.1), q.2)) :=
      (measurable_const.prodMk measurable_fst).prodMk measurable_snd
    have h := (h93 n T).2.2.1.comp hemb
    simpa only [Function.comp_def] using h
  have hm (n : ℕ) : SigmaFinite (m n T) := by
    have hcons (i : Fin (n + 1)) :
        Measurable (fun t : Fin n → ℝ => (Fin.cons (0 : ℝ) t : Fin (n + 1) → ℝ) i) := by
      induction i using Fin.cases with
      | zero => exact measurable_const
      | succ i => exact measurable_pi_apply i
    have hs : MeasurableSet (Δ n T) := by
      change MeasurableSet {t : Fin n → ℝ |
        (∀ i j : Fin (n + 1), i < j → (Fin.cons (0 : ℝ) t : Fin (n + 1) → ℝ) i < (Fin.cons (0 : ℝ) t : Fin (n + 1) → ℝ) j) ∧
        ∀ i : Fin n, t i < (T : ℝ)}
      rw [Set.ofPred_and]
      apply MeasurableSet.inter
      · simpa only [← Set.ofPred_forall] using (MeasurableSet.iInter fun (i : Fin (n + 1)) =>
          MeasurableSet.iInter fun (j : Fin (n + 1)) =>
          MeasurableSet.iInter fun (_ : i < j) => measurableSet_lt (hcons i) (hcons j))
      · simpa only [← Set.ofPred_forall] using (MeasurableSet.iInter fun (i : Fin n) =>
          measurableSet_lt (measurable_pi_apply i) (measurable_const (a := (T : ℝ))))
    have hmσ : SigmaFinite ((m n T).map (fun t : Prefix n T => t.val)) := by
      change SigmaFinite ((Measure.comap (fun t : Prefix n T => t.val)
        (volume : Measure (Fin n → ℝ))).map (fun t : Prefix n T => t.val))
      rw [map_comap_subtype_coe hs]
      infer_instance
    exact SigmaFinite.of_map (m n T) measurable_subtype_coe.aemeasurable hmσ
  have hsum : Q = Measure.sum L := by
    apply Measure.ext_of_lintegral
    intro F hF
    let D : ℕ → (E × E) → ℝ≥0∞ := fun n z =>
      ∫⁻ t : Prefix n T, F (B n (z, t)) * ρ y r z n T t ∂m n T
    have hD (n : ℕ) : Measurable (D n) := by
      letI := hm n
      exact ((hF.comp (hB n)).mul (hρ n)).lintegral_prod_right'
    have hz (z : E × E) :
        (∫⁻ w, F (z, Z y r z T w) ∂P) = ∑' n, D n z := by
      have hN : Measurable (N y r z T) := ((h93 0 T).2.2.2 y r z).2.2.1
      have hsets (n : ℕ) : MeasurableSet {w | N y r z T w = n} :=
        (measurableSet_singleton n).preimage hN
      have hdis : Pairwise (fun i j : ℕ => Disjoint {w | N y r z T w = i} {w | N y r z T w = j}) := by
        intro i j hij
        exact Set.disjoint_left.mpr (fun w hi hj => hij (hi.symm.trans hj))
      have hu : (⋃ n : ℕ, {w | N y r z T w = n}) = Set.univ := by
        ext w
        simp
      calc
        (∫⁻ w, F (z, Z y r z T w) ∂P) =
            ∑' n, ∫⁻ w in {w | N y r z T w = n}, F (z, Z y r z T w) ∂P := by
          simpa only [hu, Measure.restrict_univ] using
            (lintegral_iUnion hsets hdis (fun w => F (z, Z y r z T w)))
        _ = ∑' n, D n z := by
          apply tsum_congr
          intro n
          rcases (h93 n T).2.2.2 y r z with
            ⟨hH, hZT, hNN, hC, hGamma, hactual, hAE, hdisjoint, hcover, hlaw, hint, hmass, hbound⟩
          exact hint (fun hp => F (z, hp.2 ⟨T, le_rfl⟩))
            (hF.comp (measurable_const.prodMk ((measurable_pi_apply (⟨T, le_rfl⟩ : {s : ℝ≥0 // s ≤ T})).comp measurable_snd)))
    calc
      (∫⁻ x, F x ∂Q) = ∫⁻ z, ∫⁻ w, F (z, Z y r z T w) ∂P ∂nu y := by
        change (∫⁻ x, F x ∂Measure.map M ((nu y).prod P)) = _
        rw [lintegral_map hF hM]
        exact lintegral_prod (f := fun x => F (M x)) (hF.comp hM).aemeasurable
      _ = ∫⁻ z, ∑' n, D n z ∂nu y := lintegral_congr hz
      _ = ∑' n, ∫⁻ z, D n z ∂nu y := lintegral_tsum (fun n => (hD n).aemeasurable)
      _ = ∑' n, ∫⁻ x, F x ∂L n := by
        apply tsum_congr
        intro n
        letI := hm n
        change (∫⁻ z, ∫⁻ t : Prefix n T, F (B n (z, t)) * ρ y r z n T t ∂m n T ∂nu y) =
          ∫⁻ x, F x ∂Measure.map (B n) (sigma n)
        rw [lintegral_map hF (hB n)]
        change _ = ∫⁻ x, F (B n x) ∂(((nu y).prod (m n T)).withDensity (fun x => ρ y r x.1 n T x.2))
        rw [lintegral_withDensity_eq_lintegral_mul _
          (f := fun x : (E × E) × Prefix n T => ρ y r x.1 n T x.2)
          (g := fun x => F (B n x)) (hρ n) (hF.comp (hB n))]
        simp only [Pi.mul_apply]
        rw [lintegral_prod (f := fun x : (E × E) × Prefix n T =>
          ρ y r x.1 n T x.2 * F (B n x)) ((hρ n).mul (hF.comp (hB n))).aemeasurable]
        apply lintegral_congr
        intro z
        apply lintegral_congr
        intro t
        exact mul_comm _ _
      _ = ∫⁻ x, F x ∂Measure.sum L := (lintegral_sum_measure F L).symm
  have hsector (n : ℕ) : MeasurePreserving J (L n) (L n) :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualNormalizedEndpointSector.actual_normalized_endpoint_sector_reversal
      hα hαβ hV hH hη hβη y r n T
  refine ⟨hsum, hQ, (hsector 0).measurable, ?_⟩
  rw [hsum, Measure.map_sum (hsector 0).measurable.aemeasurable]
  congr 1
  funext n
  exact (hsector n).map_eq

#print axioms actual_countable_endpoint_law
end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualCountableEndpointLaw

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualCountableEndpointLaw.lean:159published source at ed709a7f320d

Proof architecture

Actual countable endpoint law; 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.
`simp` normalizes through registered definitional and theorem rewrites.
`apply` reduces the goal to the hypotheses of a reusable theorem.
`refine` instantiates a reusable theorem while leaving explicit subgoals.

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.
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.
A Gibbs expression is not automatically a probability law or an invariant law.