Plain-English statement
- A closed inner Pi-box is contained in a strictly larger open Pi-box. This is a bookkeeping leaf for exhaustion arguments. It only proves the coordinate set inclusion needed to feed local cutoff construction; it does not choose an exhausting sequence or construct a cutoff.
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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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
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Lean statement
theorem Icc_subset_univ_pi_Ioo_of_strict_bounds
{n : ℕ} {a b A B : Fin (n + 1) → ℝ}
(hA : ∀ i, A i < a i)
(hB : ∀ i, b i < B i) :
Set.Icc a b ⊆ Set.univ.pi (fun i => Set.Ioo (A i) (B i)) := by
intro x hx i _hi
exact ⟨lt_of_lt_of_le (hA i) (hx.1 i), lt_of_le_of_lt (hx.2 i) (hB i)⟩
/-- Smooth plateau for a finite closed Pi-box inside a strictly larger open
Pi-box.
The extra hypothesis `a ≤ b` records that the inner box is nonempty in the
intended exhaustion use. The construction comes from the generic
compact-in-open plateau theorem in `Analysis.Calculus.Cutoff`; it gives both
plain- and topological-support containment, compact support, smoothness,
`[0, 1]` range, and equality to one on the whole inner box.
This is one chosen cutoff, not yet an exhausting family with derivative
bounds. Tail passage, whole-space weighted integration by parts, generator
domains, invariant Gibbs law, reversibility, and KL/FI dissipation remain
separate obligations. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:909published source at 7bcd37294df1
Proof architecture
log-concave sampling Ch.1 cutoff/exhaustion route: an inner closed Pi-box is contained in any coordinatewise strictly larger open Pi-box
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `intro` introduces quantified hypotheses into the local proof context.
- `exact` closes the current goal with an already typed term.
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
- Pointwise support, topological support, and compact support retain distinct meanings.
- A Gibbs expression is not automatically a probability law or an invariant law.
- A formal generator display does not establish a closed operator domain.
- A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.