Plain-English statement
Every progressive product-space L2 integrand admits a sequence of bounded causal elementary adapted approximations on increasingly fine dyadic grids.
Mathematical statement
There are elementary adapted eta_n with strictly increasing dyadic levels and ||eta_n-eta||_L2 tending to zero.
Intuition
Truncation handles large values, lagged averaging preserves causality, and a fast diagonal choice makes both errors vanish together.
Conditions
- finite probability measure
- positive terminal time
- progressive product-space L2 integrand
Why these conditions cannot be dropped
- finite product measure supports dominated convergence
- positive time gives strict dyadic cells
- progressiveness is what makes lagged coefficients adapted
Proof route
- clip at increasing levels
- average each clipped process over preceding dyadic cells
- use Lebesgue differentiation and dominated convergence
- choose a fast diagonal with increasing levels
Lean interface notes
- the witness is a heterogeneous DyadicElementaryProcess sequence
- the theorem records StrictMono grid levels and actual Lp convergence
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
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.
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 progressiveL2_elementary_dense [IsFiniteMeasure mu]
(eta : ProgressiveL2Integrand filtration mu T) (hT : 0 < T) :
∃ approx : ℕ → DyadicElementaryProcess filtration T,
StrictMono (fun n ↦ (approx n).level) ∧
Tendsto (fun n ↦ (approx n).toLp mu) atTop (𝓝 eta.toLp) := by
exact ⟨canonicalElementaryApprox eta hT,
dyadicLevel_strictMono eta hT,
tendsto_canonicalElementaryApprox_toLp eta hT⟩
end ProgressiveL2Density
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheoryOpen AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:216published source at 7bcd37294df1