Plain-English statement
Real Banach/Borel Gaussian including degenerate laws and zero Lipschitz constants. Actual same-producer consumer independently verified1f746dc6. Domain only: no sharp coefficient, bias, posterior, main or cost credit.
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Proof architecture
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Lean statement
theorem integrable_and_integrable_exp_centered_of_lipschitz
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[CompleteSpace E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E]
{μ : MeasureTheory.Measure E} [ProbabilityTheory.IsGaussian μ]
{L : ℝ≥0} {f : E → ℝ} (hf : LipschitzWith L f) :
MeasureTheory.Integrable f μ ∧ ∀ t : ℝ,
MeasureTheory.Integrable
(fun x => Real.exp (t * (f x - ∫ z, f z ∂μ))) μ := by
have hbound (x : E) : |f x| ≤ (L : ℝ)*‖x‖+|f 0| := by
have hd := hf.dist_le_mul x 0
simp only [dist_eq_norm, sub_zero, Real.norm_eq_abs] at hd
calc
|f x| = |(f x-f 0)+f 0| := by congr 1; ring
_ ≤ |f x-f 0|+|f 0| := abs_add_le _ _
_ ≤ (L : ℝ)*‖x‖+|f 0| := add_le_add hd le_rfl
have hi : Integrable f μ :=
(((IsGaussian.integrable_id (μ := μ)).norm.const_mul (L : ℝ)).add
(integrable_const |f 0|)).mono' hf.continuous.aestronglyMeasurable
(ae_of_all _ fun x => by simpa only [Real.norm_eq_abs, Pi.add_apply, id_eq] using hbound x)
refine ⟨hi,fun t => ?_⟩
obtain ⟨C,hC,hint⟩ := IsGaussian.exists_integrable_exp_sq μ
let b : ℝ := |t| *(L : ℝ)
let D : ℝ := |t| *(|f 0|+|∫ z, f z ∂μ|)
have hb : 0 ≤ b := mul_nonneg (abs_nonneg _) L.coe_nonneg
have hdom (x : E) : t*(f x-∫ z, f z ∂μ) ≤
D+C⁻¹*b^2+C*‖x‖^2 := by
have hlin : t*(f x-∫ z, f z ∂μ) ≤ b*‖x‖+D := by
calc
_ ≤ |t*(f x-∫ z, f z ∂μ)| := le_abs_self _
_ = |t| *|f x-∫ z, f z ∂μ| := abs_mul _ _
_ ≤ |t| *(|f x|+|∫ z, f z ∂μ|) :=
mul_le_mul_of_nonneg_left (by simpa only [sub_zero, zero_sub, abs_neg] using
(abs_sub_le (f x) 0 (∫ z, f z ∂μ))) (abs_nonneg _)
_ ≤ |t| *((L : ℝ)*‖x‖+|f 0|+|∫ z, f z ∂μ|) :=
mul_le_mul_of_nonneg_left (add_le_add (hbound x) le_rfl) (abs_nonneg _)
_ = b*‖x‖+D := by dsimp [b,D]; ring
have hy : 2*‖x‖*b ≤ C*‖x‖^2+C⁻¹*b^2 := two_mul_le_add_mul_sq hC
have hnon : 0 ≤ b*‖x‖ := mul_nonneg hb (norm_nonneg _)
nlinarith
apply (hint.const_mul (Real.exp (D+C⁻¹*b^2))).mono'
(by fun_prop)
filter_upwards with x
rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _),← Real.exp_add]
exact Real.exp_le_exp.mpr (hdom x)
/- The zero-Lipschitz branch of the sealed sharp target is internal; this is
not an assertion of the remaining positive-dimensional concentration bound. -/Open AutoSamplingTheory/TechnicalLemmas/Probability/GaussianLipschitzExponential.lean:33published source at ed709a7f320d
Proof architecture
SPHMC actual Gaussian gradient-output centered Laplace domain; general Gaussian-observable background
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `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.
- `exact` closes the current goal with an already typed term.
Why the statement has this shape
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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.
- Almost-everywhere hypotheses depend on the stated measure and representative.