LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The type classes partition the word space and have polynomial count

Statement

Let $A$ be a finite nonempty alphabet with $k = ∣ A ∣ \geq 1$ , and let $n \geq 1$ . The type classes $T_{q}$ , $q \in P_{n} (A)$ , form a disjoint partition of $A^{n}$ , and $∣ P_{n} (A) ∣ \leq (n + 1)^{k - 1} .$

Facts & Assumptions

Given: The finite alphabet $A$ , the positive integer $n$ , and the definitions of empirical type and type class.

[F1]

Empirical types, n-types, and type classes: ${\hat{P}}_{x^{n}} (a) = N (a ∣ x^{n}) / n$ , $P_{n} (A)$ is the set of attainable empirical types, and $T_{q}$ is the set of words with empirical type $q$ .

Proof

1.1givenF1algebra

Every $x^{n} \in A^{n}$ has the single type ${\hat{P}}_{x^{n}}$ from [F1], hence lies in $T_{{\hat{P}}_{x^{n}}}$ . If $x^{n} \in T_{q} \cap T_{r}$ , its empirical vector equals both $q$ and $r$ , so $q = r$ . Thus the type classes are a disjoint cover of $A^{n}$ .

2.1F1givenalgebra∎

Fix a listing $a_{1}, \dots, a_{k}$ of $A$ . A type is determined by the first $k - 1$ counts $N (a_{i} ∣ x^{n}) \in {0, \dots, n}$ , since the final count is $n - \sum_{i < k} N (a_{i} ∣ x^{n})$ . There are at most $(n + 1)^{k - 1}$ such tuples; for $k = 1$ the tuple is empty and determines the unique type.

Depends on

Empirical types, n-types, and type classes

Used by

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, author-hosted book draft (standard reference, not scraped)
Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, second edition, Chapter 11 (standard reference, not scraped)