actual_l2_strong_continuity

Plain-English statement

- Zero-time strong continuity for every real L2 observable of the same actual PBPS phase process, retaining its previously proved flip and averaging laws. This is only the zero-time component of Proposition A.1(ii), not a semigroup law.

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_l2_strong_continuity
    {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_l2_strong_continuity_statement hα hαβ hV hH hη hβη := by
  classical
  unfold actual_l2_strong_continuity_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, h106, hBorel, h109⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPositionL2Averaging.actual_position_l2_averaging
      hα hαβ hV hH hη hβη
  obtain ⟨Zb, hZbm, hZba, hZbf, hZbAE, hshort, htest, houter⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualOuterBoundedL2Continuity.actual_outer_bounded_l2_continuity
      hα hαβ hV hH hη hβη
  have hZeq : Zb = Z := by
    funext y xRef z t w
    by_cases hx : ∃ n, w ∈ I y xRef z t n
    · obtain ⟨n, hn⟩ := hx
      obtain ⟨a, ha⟩ : ∃ a, record y xRef z (ε w) n = Sum.inl a := by
        cases hr : record y xRef z (ε w) n with
        | inl a => exact ⟨a, rfl⟩
        | inr u =>
          have hh := hn.1
          change (record y xRef z (ε w) n).elim
            (fun a => (a.1 : WithTop ℝ≥0)) (fun _ => ⊤) ≤ (t : WithTop ℝ≥0) at hh
          rw [hr] at hh
          exact False.elim ((WithTop.not_top_le_coe t) hh)
      exact (hZba y xRef z w t n a hn.1 hn.2 ha).trans
        (hZa y xRef z w t n a hn ha).symm
    · exact (hZbf y xRef z w t hx).trans (hZf y xRef z w t hx).symm
  rw [hZeq] at houter
  refine ⟨Z, hZm, hZa, hZf, hZ0, h106, hBorel, ?_⟩
  intro y xRef
  obtain ⟨L, hL0, hL, hR, hJ⟩ := h109 y xRef
  refine ⟨L, hL0, ?_, hL, hR, hJ⟩
  have hnu :
      ((volume : Measure E).tilted (fun x => -V x - ‖x - y‖ ^ 2 / (2 * η))).prod
        (stdGaussian E) = nu y := by
    apply congrArg (fun μ : Measure E => μ.prod (stdGaussian E))
    apply congrArg ((volume : Measure E).tilted)
    funext x
    dsimp only [Vy]
    ring
  letI : IsProbabilityMeasure (nu y) := by
    rw [← hnu]
    exact (houter.1 y).2
  have hcore (g : (E × E) →ᵇ ℝ) :
      Tendsto (fun t : ℝ≥0 => L t (BoundedContinuousFunction.toLp 2 (nu y) ℝ g))
        (𝓝 0) (𝓝 (BoundedContinuousFunction.toLp 2 (nu y) ℝ g)) := by
    let f : Lp ℝ 2 (nu y) := BoundedContinuousFunction.toLp 2 (nu y) ℝ g
    have hf : (f : (E × E) → ℝ) =ᵐ[nu y] g :=
      BoundedContinuousFunction.coeFn_toLp 2 (nu y) ℝ g
    have hg : MemLp g 2 (nu y) := MemLp.ae_eq hf (Lp.memLp f)
    have hfg : hg.toLp g = f := by
      apply Lp.ext
      exact hg.coeFn_toLp.trans hf.symm
    have hlimit := (houter.2 y xRef g g.continuous ‖g‖ (norm_nonneg g)
      (fun z => by simpa only [Real.norm_eq_abs] using g.norm_coe_le_norm z)).2
    dsimp only at hlimit
    rw [hnu] at hlimit
    have hnorm (t : ℝ≥0) :
        ‖L t f - f‖ ^ 2 =
          ∫ z, ((∫ w, g (Z y xRef z t w) ∂P) - g z) ^ 2 ∂nu y := by
      obtain ⟨_, ⟨hop, he⟩, _⟩ := (hL t).2 g hg
      rw [hfg] at he
      have ha : (L t f : (E × E) → ℝ) =ᵐ[nu y]
          (fun z => ∫ w, g (Z y xRef z t w) ∂P) := by
        rw [he]
        exact hop.coeFn_toLp
      rw [← real_inner_self_eq_norm_sq (L t f - f), L2.inner_def]
      apply integral_congr_ae
      filter_upwards [Lp.coeFn_sub (L t f) f, ha, hf] with z hz hzA hzf
      simp only [hz, Pi.sub_apply, hzA, hzf, RCLike.inner_apply, conj_trivial, pow_two]
    have hnormlimit : Tendsto (fun t : ℝ≥0 => ‖L t f - f‖ ^ 2) (𝓝 0) (𝓝 0) := by
      simpa only [hnorm] using hlimit
    have hsqrt := (Real.continuous_sqrt.tendsto 0).comp hnormlimit
    change Tendsto (fun t : ℝ≥0 => Real.sqrt (‖L t f - f‖ ^ 2))
      (𝓝 0) (𝓝 (Real.sqrt 0)) at hsqrt
    have hzero : Tendsto (fun t : ℝ≥0 => ‖L t f - f‖) (𝓝 0) (𝓝 0) := by
      simpa only [Real.sqrt_sq_eq_abs, abs_norm, Real.sqrt_zero] using hsqrt
    exact tendsto_iff_norm_sub_tendsto_zero.mpr hzero
  have hequi : Equicontinuous (fun t : ℝ≥0 => fun f : Lp ℝ 2 (nu y) => L t f) := by
    apply Metric.equicontinuous_of_continuity_modulus id tendsto_id
    intro f g t
    have hlip : LipschitzWith 1 (L t) :=
      ContinuousLinearMap.lipschitzWith_of_opNorm_le (hL t).1
    simpa only [id_eq, NNReal.coe_one, one_mul] using hlip.dist_le_mul f g
  have hclosed : IsClosed {f : Lp ℝ 2 (nu y) |
      Tendsto (fun t : ℝ≥0 => L t f) (𝓝 0) (𝓝 f)} :=
    hequi.isClosed_setOfPred_tendsto continuous_id
  have hdense : DenseRange (BoundedContinuousFunction.toLp 2 (nu y) ℝ :
      ((E × E) →ᵇ ℝ) →L[ℝ] Lp ℝ 2 (nu y)) :=
    BoundedContinuousFunction.toLp_denseRange ℝ (nu y) ℝ (by norm_num)
  have hsubset : Set.range (BoundedContinuousFunction.toLp 2 (nu y) ℝ :
      ((E × E) →ᵇ ℝ) →L[ℝ] Lp ℝ 2 (nu y)) ⊆
      {f : Lp ℝ 2 (nu y) | Tendsto (fun t : ℝ≥0 => L t f) (𝓝 0) (𝓝 f)} := by
    rintro _ ⟨g, rfl⟩
    exact hcore g
  intro f
  exact closure_minimal hsubset hclosed (hdense f)

#print axioms actual_l2_strong_continuity
end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualL2StrongContinuity

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualL2StrongContinuity.lean:212published source at ed709a7f320d

Proof architecture

Actual same-process PBPS semigroup prerequisite; 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.
`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.
`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.
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.