CoreContract · Samplinglib

Plain-English statement

A candidate Langevin generator must declare a function domain containing every compactly supported C2 test and must agree with the displayed Langevin differential operator on that core.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Mathematical statement

C_c^2 is contained in D(A), and Af = Delta f - <grad V, grad f> for every f in C_c^2.

Intuition

Writing down a differential formula does not determine the domain of an unbounded generator. The contract keeps domain membership and formula agreement as separate proof obligations.

Conditions

A candidate operator action and an explicit domain set are supplied.
The state space is finite-dimensional EuclideanSpace in the current interface.

Why these conditions cannot be dropped

Unbounded operators with the same pointwise formula can have different domains and closures.
The current IBP theorem is established only on the compactly supported C2 core.

Proof route

Name the compactly supported C2 predicate.
Name the displayed Langevin operator independently.
Record core inclusion and operator agreement as fields that a concrete semigroup generator must discharge.

Lean interface notes

CoreContract is a Prop-valued structure, so it packages evidence rather than constructing an operator.
It does not assert closedness, closability, strong continuity, or semigroup generation.

Lean learning studio · mathematics → formal proof

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.

BeginnerWhy this theorem exists → intuition → statement → one hand calculation. Hide proof-engineering detail.

RigorousExpose assumptions, hidden measure/limit/domain issues, proof route, and rigorous references.

Lean learnerOpen the exact declaration, proof tree/network, syntax glossary, and line-by-line explanation.

Proof architecture

Start with the tree when learning: prerequisites sit below the theorem and downstream results sit above it. Switch to the network when you want to understand where this declaration lives in the local formal library. Every mapped node is clickable.

Loading source-derived dependency evidence…

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

structure CoreContract
    {n : ℕ}
    (V : EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
    (generator :
      (EuclideanSpace ℝ (Fin (n + 1)) → ℝ) →
        EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
    (domain : Set (EuclideanSpace ℝ (Fin (n + 1)) → ℝ)) : Prop where
  core_mem_domain : ∀ f, CompactlySupportedC2 f → f ∈ domain
  generator_eq_operator_on_core :
    ∀ f, CompactlySupportedC2 f → generator f = operator V f

/-- The normalized Gibbs measure annihilates the displayed Langevin operator
on the compactly supported `C²` core.

This is a normalized-measure corollary of the whole-space weighted-IBP theorem.
It is a core-level infinitesimal stationarity statement, not semigroup
invariance: extending it to a semigroup-stable generator domain requires an
additional closure/core theorem or a separate martingale-problem argument. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinGenerator.lean:47published source at 7bcd37294df1