Plain-English statement
- Accumulated energy is continuous for almost every sample point.
Read the mathematics first, then descend into Lean
You do not need to know Lean before opening this panel. The page keeps the paper-level theorem, rigorous proof obligations, exact Lean declaration, and proof dependencies as separate layers so a first-time reader can move down one layer at a time.
Proof architecture
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Graph rule: only dependencies found by the ASTIS source scan are drawn. Missing tactic indirection is treated as an under-approximation; the site never invents an edge just to make a prettier graph.
How to read the exact Lean declaration
Read a Lean theorem left-to-right exactly as you would unpack a mathematical sentence: name → ambient types → automatically inferred structures → explicit hypotheses → conclusion → proof. Then read the proof top-to-bottom as transformations of the current goal.
- NameWhat reusable mathematical fact is being created?
- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
- PropositionAfter the colon, translate the Lean expression back into a paper statement.
- Proof actionsAfter
by, ask what each tactic does to the mathematical goal—not only what syntax it uses.
Syntax used on this page
This glossary is filtered to syntax that actually occurs in the declaration above. Open a symbol only when you meet it, rather than memorizing Lean grammar in advance.
Source voice and ASTIS voice stay separate
When Samplinglib shows a short quotation from Chewi, it is labeled as a source excerpt and linked to the canonical book page. Intuition, expanded proof steps, hidden regularity assumptions, and Lean explanations are ASTIS-authored commentary. A quotation never substitutes for a formal proof, and an ASTIS explanation is never attributed to the textbook author.
Lean statement
theorem continuous_accumulatedEnergyReal_ae
(eta : LocalProgressiveL2Integrand filtration mu T) :
∀ᵐ omega ∂mu,
Continuous (fun t => accumulatedEnergyReal eta t omega) := by
filter_upwards [sectionSquare_integrable_ae eta] with omega homega
exact continuous_accumulatedEnergyReal_of_integrable eta omega homega
@[simp] theorem accumulatedEnergyReal_zero
(eta : LocalProgressiveL2Integrand filtration mu T) (omega : Omega) :
accumulatedEnergyReal eta 0 omega = 0 := by
rw [accumulatedEnergyReal_eq_prefixIntegral]
exact prefixIntegral_zero _ _
/-- On every finite-energy path, accumulated energy is monotone. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyPathContinuity.lean:99published source at 7bcd37294df1
Proof architecture
justify the first-hitting canonical localizer by a continuous monotone energy path
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `rw` rewrites by an established identity.
- `simp` normalizes through registered definitional and theorem rewrites.
- `exact` closes the current goal with an already typed term.
- `filter_upwards` moves an almost-everywhere or eventual statement into a pointwise local context.
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.
- Almost-everywhere hypotheses depend on the stated measure and representative.