ExampleConstruction: AI-adaptedVerification: 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.

Binary type classes at length ten

Example

For $A = {0, 1}$ and $n = 10$ , the types with $q (1) = 1 / 5$ and $q (1) = 1 / 2$ have class sizes $45$ and $252$ . Their entropy exponentials are respectively $5^{10} / 4^{8}$ and $2^{10} = 1024$ , and the polynomially corrected bounds use the factor $(n + 1)^{- 1} = 1 / 11$ .

Facts & Assumptions

Given: The binary alphabet $A = {0, 1}$ , block length $n = 10$ , and the two stated attainable types.

[F1]

Empirical types, n-types, and type classes: a type class consists of the words with prescribed symbol counts.

[F2]

Finite distributions, entropy, divergence, and stochastic matrices: $H (q) = - \sum_{a : q (a) > 0} q (a) \ln q (a)$ in nats.

[F3]

Polynomially sharp bounds for type-class cardinality: for an alphabet of size $k$ , $(n + 1)^{- (k - 1)} e^{n H (q)} \leq ∣ T_{q} ∣ \leq e^{n H (q)}$ .

Verification

Given: The data above.

1.1F1F2givenalgebra

For $q (1) = 1 / 5$ , the counts are $(8, 2)$ , so $∣ T_{q} ∣ = 10! / (8! 2!) = 45$ and $e^{10 H (q)} = (1 / 5)^{- 2} (4 / 5)^{- 8} = 5^{10} / 4^{8}$ .

1.2F1F2F3givenalgebra

For $q (1) = 1 / 2$ , the counts are $(5, 5)$ , so $∣ T_{q} ∣ = 10! / (5! 5!) = 252$ and $e^{10 H (q)} = 2^{10} = 1024$ . Since $1024 \leq 252 \cdot 11 = 2,772$ and $252 \leq 1,024$ , we have $1024 / 11 \leq 252 \leq 1024$ , retaining the polynomial factor in [F3].

2.1F3step 1.1algebra

Since $5^{10} = 9,765,625$ , $4^{8} = 65,536$ , $45 \cdot 11 \cdot 4^{8} = 32,440,320$ , and $45 \cdot 4^{8} = 2,949,120$ , cross-multiplication in step 1.1 gives $e^{10 H (q)} / 11 \leq 45 \leq e^{10 H (q)}$ , as [F3] states.

3.1F1F2F3algebra∎

At endpoint types $q = (1, 0)$ and $q = (0, 1)$ , each type class is a singleton and $H (q) = 0$ , so the lower cardinality bound is $1 / 11$ , while the class size and upper bound equal $1$ ; thus [F3] also covers zero-coordinate boundary cases.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

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