DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Finite-alphabet IID multisource

Definition

Fix an integer $k \geq 2$ and finite nonempty alphabets $X_{1}, \dots, X_{k}$ . Let $X : = \prod_{i = 1}^{k} X_{i}$ , identified with a finite alphabet as in Finite-alphabet IID source. A finite-alphabet IID $k$ -source is an IID sequence of vectors $Z_{t} = (X_{1, t}, \dots, X_{k, t}) \in X, t \geq 1,$ with common joint probability vector $P$ on $X$ . Thus for every $n \geq 1$ its block law is $P^{\otimes n}$ , and terminal $i$ observes $X_{i}^{n} = (X_{i, 1}, \dots, X_{i, n})$ . For $S \subseteq [k] : = {1, \dots, k}$ write $X_{S} = (X_{i})_{i \in S}$ , let $S^{c} = [k] ∖ S$ , and define $H_{P} (X_{S} ∣ X_{S^{c}})$ by Finite conditional entropy for the induced finite joint law. When $S = [k]$ , $X_{S^{c}}$ takes values in the singleton empty product, so $H_{P} (X_{S} ∣ X_{S^{c}}) = H (P)$ under Finite distributions, entropy, divergence, and stochastic matrices. Unmarked logarithms and information rates use nats by Information units and logarithm-base convention.

Depends on

Used by

Finite fixed-length multiterminal lossless code Definition
Finite-alphabet lossy source coding with decoder-only side information Definition
Finite-alphabet multiterminal lossy source code Definition
Jointly typical blocks for a finite IID multisource Definition
Lossless transmission of two correlated IID sources over a DM-MAC Definition
Berger-Tung inner bound contains the Slepian-Wolf region Example
Binary symmetric source: Slepian–Wolf corners and common part Example
Lossless Wyner-Ziv endpoint is the Slepian-Wolf rate Example
Shared random bins do not factor across encoders Example
Conditional block law factors across IID coordinates Lemma
Joint typicality probability tends to one for a finite IID multisource Lemma
Finite-alphabet Berger-Tung multiterminal lossy inner bound Theorem
Finite-alphabet memoryless Wyner-Ziv theorem Theorem
Finite-IID k-source Slepian–Wolf rate region Theorem

Dependency tree · two levels

12 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 15 (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, author-hosted book draft (standard reference, not scraped)