Plain-English statement
- Energy form of the one-sided DV bound for a scaled selected test. For SALD use sites, `q` is a squared velocity or residual norm. This theorem starts after the selected-test hypotheses have been supplied, applies the compiled one-sided backend for `Z=alpha*q`, divides by `alpha > 0`, and rewrites the log-mgf quotient as the supplied alpha-complexity density `eAlpha`. The Boucheron supremum equality from `appendix.tex:73-79` remains source-cited.
Read the mathematics first, then descend into Lean
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Proof architecture
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How to read the exact Lean declaration
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- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
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Syntax used on this page
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Lean statement
theorem dvVariationalScaledTestEnergyBound {Ω : Type*} [MeasurableSpace Ω]
(nu mu : Measure Ω) [IsProbabilityMeasure nu] [IsProbabilityMeasure mu]
[SigmaFinite mu] [SigmaFinite nu]
(q : Ω → Real) {alpha alpha0 eAlpha : Real}
(hAlpha_pos : 0 < alpha) (hAlpha_le : alpha ≤ alpha0)
(hnu_mu : nu ≪ mu)
(hq_nu : Integrable q nu)
(hexp_alpha0_mu : Integrable (fun x ↦ Real.exp (alpha0 * q x)) mu)
(hllr : Integrable (llr nu mu) nu)
(heAlpha : eAlpha =
alpha⁻¹ * Real.log (∫ x, Real.exp (alpha * q x) ∂mu)) :
(∫ x, q x ∂nu) ≤ alpha⁻¹ * (klDiv nu mu).toReal + eAlpha := by
have hZ_nu : Integrable (fun x ↦ alpha * q x) nu := by
simpa [smul_eq_mul] using hq_nu.const_mul alpha
have hdv := dvVariationalOneSidedOfScaledTest (nu := nu) (mu := mu)
(q := q) hAlpha_pos.le hAlpha_le hnu_mu hZ_nu hexp_alpha0_mu hllr
have hscaledIntegral :
(∫ x, alpha * q x ∂nu) = alpha * ∫ x, q x ∂nu := by
rw [integral_const_mul]
have hscaled :
alpha * (∫ x, q x ∂nu) ≤
(klDiv nu mu).toReal + Real.log (∫ x, Real.exp (alpha * q x) ∂mu) := by
have hvar :
alpha * (∫ x, q x ∂nu) -
Real.log (∫ x, Real.exp (alpha * q x) ∂mu) ≤
(klDiv nu mu).toReal := by
simpa [hscaledIntegral] using hdv
exact sub_le_iff_le_add.mp hvar
have hdiv :
alpha⁻¹ * (alpha * (∫ x, q x ∂nu)) ≤
alpha⁻¹ * ((klDiv nu mu).toReal +
Real.log (∫ x, Real.exp (alpha * q x) ∂mu)) := by
exact mul_le_mul_of_nonneg_left hscaled (inv_nonneg.mpr hAlpha_pos.le)
calc
(∫ x, q x ∂nu) = alpha⁻¹ * (alpha * (∫ x, q x ∂nu)) := by
field_simp [ne_of_gt hAlpha_pos]
_ ≤ alpha⁻¹ * ((klDiv nu mu).toReal +
Real.log (∫ x, Real.exp (alpha * q x) ∂mu)) := hdiv
_ = alpha⁻¹ * (klDiv nu mu).toReal + eAlpha := by
rw [heAlpha]
ring
/-- Coefficient-preserving energy form of the selected scaled-test DV bound.
This is the local algebraic shape used before Gronwall in SALD proofs after a
nonnegative prefactor, such as `(1/2)*dot{s}(t)^(-1)`, multiplies the
post-DV energy estimate.
-/Proof architecture
convert finite log-mgf and KL hypotheses into residual energy bounds
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.
- `exact` closes the current goal with an already typed term.
- `simpa` closes the goal after a controlled simplification of a typed result.
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.