gaussian_reflected_mean_c1

Plain-English statement

- Compact C1 observables have C1 means under the literal reflected Gaussian posterior. No input moment, density or analytic certificate is assumed.

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 gaussian_reflected_mean_c1
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    (μ : Measure E) [IsProbabilityMeasure μ] {η : ℝ} (hη : 0 < η)
    {f : E → ℝ} (hf : ContDiff ℝ 1 f) (hc : HasCompactSupport f) :
    let R := fun y : E => μ.tilted (fun x => -‖x-y‖^2/(2*η))
    let S := fun y : E => (R y).map (fun x : E => (2:ℝ) • x-y)
    ContDiff ℝ 1 (fun y : E => ∫ u, f u ∂S y)
 := by
  let K := fun y x : E => Real.exp (-‖y-x‖^2/(2*η))
  let K' := fun y x : E => (-K y x/η) • innerSL ℝ (y-x)
  let F := fun y x : E => K y x * f ((2:ℝ) • x-y)
  let G := fun y x : E => K y x • (-fderiv ℝ f ((2:ℝ) • x-y)) +
    f ((2:ℝ) • x-y) • K' y x
  let N := fun y : E => ∫ x, F y x ∂μ
  let D := fun y : E => ∫ x, G y x ∂μ
  let Z := fun y : E => ∫ x, K y x ∂μ
  have hdfC : Continuous (fderiv ℝ f) := hf.continuous_fderiv one_ne_zero
  obtain ⟨Mf, hMf⟩ := hc.exists_bound_of_continuous hf.continuous
  obtain ⟨Md, hMd⟩ := (hc.fderiv ℝ).exists_bound_of_continuous hdfC
  have hMf0 : 0 ≤ Mf := (norm_nonneg (f 0)).trans (hMf 0)
  have hK (y x : E) : 0 < K y x ∧ K y x ≤ 1 ∧ K y x * ‖y-x‖^2 ≤ 2*η := by
    have hp : 0 < K y x := Real.exp_pos _
    have h1 : K y x ≤ 1 := Real.exp_le_one_iff.mpr
      (div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (sq_nonneg _)) (by positivity))
    have hb := Real.mul_exp_neg_le_exp_neg_one (‖y-x‖^2/(2*η))
    have he : Real.exp (-1) ≤ 1 := Real.exp_le_one_iff.mpr (by norm_num)
    have hb' : K y x * (‖y-x‖^2/(2*η)) ≤ 1 := by
      simpa [K, mul_comm, neg_div] using hb.trans he
    have heq : K y x * (‖y-x‖^2/(2*η)) = (K y x * ‖y-x‖^2)/(2*η) := by ring
    rw [heq, div_le_iff₀ (by positivity)] at hb'
    exact ⟨hp,h1,by simpa using hb'⟩
  have hKderiv (y x : E) : HasFDerivAt (fun z => K z x) (K' y x) y := by
    have hd := (((hasFDerivAt_id y).sub_const x).norm_sq.const_mul (-1/(2*η))).exp
    convert hd using 1
    · funext z; dsimp only [K,id]; congr 1; ring
    · ext v
      have he : -‖y-x‖^2/(2*η) = (-1/(2*η))*‖y-x‖^2 := by ring
      simp only [K',K,he,smul_apply,ContinuousLinearMap.comp_apply,
        ContinuousLinearMap.id_apply,smul_eq_mul,innerSL_apply_apply,id]
      ring
  have hKbound (y x : E) : ‖K' y x‖ ≤ (1+2*η)/η := by
    obtain ⟨hp,h1,h2⟩ := hK y x
    have hr : ‖y-x‖ ≤ 1+‖y-x‖^2 := by nlinarith [sq_nonneg (‖y-x‖-1)]
    have hb : K y x * ‖y-x‖ ≤ 1+2*η := by
      calc
        _ ≤ K y x * (1+‖y-x‖^2) := mul_le_mul_of_nonneg_left hr hp.le
        _ ≤ 1+2*η := by nlinarith
    dsimp only [K']
    rw [norm_smul,Real.norm_eq_abs,abs_div,abs_neg,abs_of_pos hp,
      abs_of_pos hη,innerSL_apply_norm,div_mul_eq_mul_div,div_le_div_iff_of_pos_right hη]
    exact hb
  have hGbound (y x : E) : ‖G y x‖ ≤ Md + Mf*((1+2*η)/η) := by
    have hk := hK y x
    calc
      _ ≤ ‖K y x • (-fderiv ℝ f ((2:ℝ) • x-y))‖ +
          ‖f ((2:ℝ) • x-y) • K' y x‖ := norm_add_le _ _
      _ = K y x * ‖fderiv ℝ f ((2:ℝ) • x-y)‖ +
          ‖f ((2:ℝ) • x-y)‖ * ‖K' y x‖ := by
            rw [norm_smul,norm_smul,norm_neg,Real.norm_eq_abs,abs_of_pos hk.1]
      _ ≤ 1*Md + Mf*((1+2*η)/η) := add_le_add
        (mul_le_mul hk.2.1 (hMd _) (norm_nonneg _) (by norm_num))
        (mul_le_mul (hMf _) (hKbound y x) (norm_nonneg _) hMf0)
      _ = _ := by ring
  have hFmeas (y : E) : AEStronglyMeasurable (F y) μ := by
    apply Continuous.aestronglyMeasurable
    have hk : Continuous (K y) := by unfold K; fun_prop
    exact hk.mul (hf.continuous.comp (by fun_prop))
  have hGmeas (y : E) : AEStronglyMeasurable (G y) μ := by
    apply Continuous.aestronglyMeasurable
    dsimp only [G,K',K]
    have hd : Continuous (fun x : E => fderiv ℝ f ((2:ℝ) • x-y)) :=
      hdfC.comp (by fun_prop)
    have ho : Continuous (fun x : E => f ((2:ℝ) • x-y)) := hf.continuous.comp (by fun_prop)
    fun_prop
  have hFI (y : E) : Integrable (F y) μ := by
    apply (integrable_const Mf).mono' (hFmeas y)
    filter_upwards with x
    dsimp only [F]
    rw [norm_mul,Real.norm_eq_abs,abs_of_pos (hK y x).1]
    calc
      _ ≤ 1*Mf := mul_le_mul (hK y x).2.1 (hMf _) (norm_nonneg _) (by norm_num)
      _ = _ := one_mul _
  have hFderiv (y x : E) : HasFDerivAt (fun z => F z x) (G y x) y := by
    have ha := (hasFDerivAt_id (𝕜 := ℝ) y).const_sub ((2:ℝ) • x)
    have ho := (hf.differentiable_one ((2:ℝ) • x-y)).hasFDerivAt.comp y ha
    have ho' : HasFDerivAt (fun z : E => f ((2:ℝ) • x-z))
        (-fderiv ℝ f ((2:ℝ) • x-y)) y := by
      convert ho using 1 <;> first | rfl | (ext v; simp)
    exact (hKderiv y x).mul ho'
  have hdN (y : E) : HasFDerivAt N (D y) y := by
    apply hasFDerivAt_integral_of_dominated_of_fderiv_le (s := Set.univ)
      (bound := fun _ => Md+Mf*((1+2*η)/η)) Filter.univ_mem
    · exact Filter.Eventually.of_forall hFmeas
    · exact hFI y
    · exact hGmeas y
    · exact Filter.Eventually.of_forall (fun x z _ => hGbound z x)
    · exact integrable_const _
    · exact Filter.Eventually.of_forall (fun x z _ => hFderiv z x)
  have hDC : Continuous D := by
    apply continuous_of_dominated hGmeas
      (bound := fun _ => Md+Mf*((1+2*η)/η))
      (fun y => Filter.Eventually.of_forall (fun x => hGbound y x)) (integrable_const _)
    filter_upwards with x
    dsimp only [G,K',K]
    have hd : Continuous (fun y : E => fderiv ℝ f ((2:ℝ) • x-y)) := hdfC.comp (by fun_prop)
    have ho : Continuous (fun y : E => f ((2:ℝ) • x-y)) := hf.continuous.comp (by fun_prop)
    fun_prop
  have hNC : ContDiff ℝ 1 N := contDiff_one_iff_hasFDerivAt.mpr ⟨D,hDC,hdN⟩
  have hparent := GaussianConvolutionRegularity.gaussian_convolution_potential_c2 μ hη
  have hZC : ContDiff ℝ 2 Z := hparent.2.2.1
  have hZ (y : E) : Z y ≠ 0 := by
    have hp := hparent.2.1 y
    intro hz
    change 0 < ((Real.sqrt (2*Real.pi*η))⁻¹)^Module.finrank ℝ E * Z y at hp
    rw [hz,mul_zero] at hp
    exact lt_irrefl _ hp
  have hKI (y : E) : Integrable (K y) μ := by
    apply (integrable_const (1:ℝ)).mono' (by unfold K; fun_prop)
    filter_upwards with x
    rw [Real.norm_eq_abs,abs_of_pos (hK y x).1]
    exact (hK y x).2.1
  have hmean (y : E) :
      (∫ u, f u ∂((μ.tilted (fun x => -‖x-y‖^2/(2*η))).map
        (fun x : E => (2:ℝ) • x-y))) = N y / Z y := by
    have hlike : Integrable (fun x : E => Real.exp (-‖x-y‖^2/(2*η))) μ := by
      simpa only [K,norm_sub_rev] using hKI y
    have : IsProbabilityMeasure (μ.tilted (fun x => -‖x-y‖^2/(2*η))) :=
      isProbabilityMeasure_tilted hlike
    have hobs : Integrable (fun x : E => f ((2:ℝ) • x-y))
        (μ.tilted (fun x => -‖x-y‖^2/(2*η))) := by
      apply (integrable_const Mf).mono' (hf.continuous.comp (by fun_prop)).aestronglyMeasurable
      exact Filter.Eventually.of_forall (fun x => hMf _)
    have hmapI : Integrable f ((μ.tilted (fun x => -‖x-y‖^2/(2*η))).map
        (fun x : E => (2:ℝ) • x-y)) :=
      (integrable_map_measure hf.continuous.aestronglyMeasurable (by fun_prop)).mpr hobs
    rw [integral_map (by fun_prop) hmapI.aestronglyMeasurable,integral_tilted]
    have hw (x : E) : Real.exp (-‖x-y‖^2/(2*η)) = K y x := by
      dsimp [K]; rw [norm_sub_rev x y]
    simp_rw [hw,smul_eq_mul]
    rw [← integral_div]
    apply integral_congr_ae
    filter_upwards with x
    dsimp [N,Z,F]
    ring
  change ContDiff ℝ 1 (fun y : E => ∫ u, f u ∂((μ.tilted
    (fun x => -‖x-y‖^2/(2*η))).map (fun x : E => (2:ℝ) • x-y)))
  simp_rw [hmean]
  exact hNC.div (hZC.of_le (by norm_num)) hZ

end
end AutoSamplingTheory.TechnicalLemmas.Measure.GaussianReflectedMean

Open AutoSamplingTheory/TechnicalLemmas/Measure/GaussianReflectedMean.lean:28published source at ed709a7f320d

Proof architecture

Shared Gaussian posterior compact-observer regularity; no SALD admission

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.
`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.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.
Pointwise support, topological support, and compact support retain distinct meanings.
Almost-everywhere hypotheses depend on the stated measure and representative.
Density statements retain normalization and absolute-continuity prerequisites.