Plain-English statement
- Whole-space integrability of the Gibbs-weighted coordinate derivative field from finiteness of the unnormalized Gibbs mass and a uniform operator norm bound on the test-function derivative. This theorem proves only source-field integrability. It does not prove a cutoff main-term limit, weighted integration by parts, stationarity, or an invariant Gibbs law.
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Proof architecture
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Lean statement
theorem integrable_expNeg_fderivCoordinateField_of_lintegral_expNeg_ne_top_of_fderiv_norm_le
{n : ℕ}
{V f : EuclideanSpace ℝ (Fin (n + 1)) → ℝ} {C : ℝ}
(hV : Continuous V)
(hf : ContDiff ℝ 1 f)
(hZ : (∫⁻ y : EuclideanSpace ℝ (Fin (n + 1)),
ENNReal.ofReal (Real.exp (-V y)) ∂volume) ≠ ∞)
(hf_bound : ∀ y, ‖fderiv ℝ f y‖ ≤ C) :
Integrable
(fun x : Fin (n + 1) → ℝ => fun i =>
Real.exp (-V (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))) *
fderiv ℝ f (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))
(EuclideanSpace.single i (1 : ℝ))) volume := by
have hweight :=
integrable_expNeg_comp_toLp_of_lintegral_expNeg_ne_top hV hZ
let D : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) :=
fun x i =>
fderiv ℝ f (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))
(EuclideanSpace.single i (1 : ℝ))
have hD_continuous : Continuous D := by
refine continuous_pi fun i => ?_
exact
((hf.continuous_fderiv one_ne_zero).comp
(PiLp.continuous_toLp 2 _)).clm_apply continuous_const
have hC : 0 ≤ C := by
exact (norm_nonneg (fderiv ℝ f 0)).trans (hf_bound 0)
have hD_bound : ∀ x, ‖D x‖ ≤ C := by
intro x
refine (pi_norm_le_iff_of_nonneg hC).2 fun i => ?_
calc
‖D x i‖ ≤
‖fderiv ℝ f
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))‖ *
‖EuclideanSpace.single i (1 : ℝ)‖ :=
ContinuousLinearMap.le_opNorm _ _
_ = ‖fderiv ℝ f
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))‖ := by
rw [(EuclideanSpace.orthonormal_single (𝕜 := ℝ)).1 i]
simp
_ ≤ C := hf_bound _
have hproduct := hweight.smul_bdd C hD_continuous.aestronglyMeasurable
(Filter.Eventually.of_forall hD_bound)
simpa [D, Pi.smul_apply, smul_eq_mul] using! hproduct
end Langevin
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheoryOpen AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1253published source at 7bcd37294df1
Proof architecture
Chewi Ch.1 Example 1.2.8 cutoff route: discharge the concrete `Integrable G volume` premise for `G = exp(-V) * fderiv f` in raw Pi coordinates from finite Gibbs mass and a bounded first derivative
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.
- `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.
- A Gibbs expression is not automatically a probability law or an invariant law.
- A formal generator display does not establish a closed operator domain.
- A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.