Plain-English statement
- The fully expanded diagonal `CD(alpha, infinity)` inequality gives the generator expression used in backward interpolation. To keep the public interface directly inspectable, the statement exposes positivity of `alpha`, every curvature quantifier, and the exact one-half normalization of the iterated carré du champ in its signature.
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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- NameWhat reusable mathematical fact is being created?
- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
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- Proof actionsAfter
by, ask what each tactic does to the mathematical goal—not only what syntax it uses.
Syntax used on this page
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Lean statement
theorem interpolationDerivative_lower_bound_of_expandedBakryEmery
(generator : (X → ℝ) →ₗ[ℝ] (X → ℝ))
{alpha : ℝ} (_halpha : 0 < alpha)
(hcurvature : ∀ (g : X → ℝ) (y : X),
alpha * carreDuChamp generator g g y ≤
(2 : ℝ)⁻¹ *
(generator (carreDuChamp generator g g) y -
carreDuChamp generator g (generator g) y -
carreDuChamp generator g (generator g) y))
(f : X → ℝ) (x : X) :
2 * alpha * carreDuChamp generator f f x ≤
generator (carreDuChamp generator f f) x -
2 * carreDuChamp generator f (generator f) x := by
have h := hcurvature f x
nlinarith
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheoryOpen AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BakryEmeryCalculus.lean:30published source at ed709a7f320d
Proof architecture
derive the exact lower bound for the backward interpolation derivative expression from CD(alpha, infinity)
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.
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
- A formal generator display does not establish a closed operator domain.