DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

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 binary source codes, extension, and expected length

Definition

A binary word is a finite sequence of symbols from ${0, 1}$ . The set ${0, 1}^{*}$ includes the empty word $ε$ , and ${0, 1}^{+}$ is the set of nonempty binary words. A binary source code for the finite source $X = [m]$ is a function $C : X \to {0, 1}^{+}$ . Its codeword length is $ℓ_{i} : = ∣ C (i) ∣ \geq 1$ .

The extension $C^{*} : X^{*} \to {0, 1}^{*}$ maps the empty source word to $ε$ and maps $i_{1} \dots i_{n}$ to $C (i_{1}) \dots C (i_{n})$ . The code is uniquely decodable if $C^{*}$ is injective, including comparisons between source words of different lengths. Its expected length for source law $p$ is $L_{p} (C) : = \sum_{i = 1}^{m} p_{i} ℓ_{i} .$

Depends on

Finite discrete sources and product blocks

Used by

Deterministic binary Huffman coding Algorithm
Exact finite-register arithmetic coding for rational block sources Algorithm
Binary prefix codes and their code trees Definition
Ideal-real arithmetic interval coding for a fixed block Definition
A uniquely decodable binary code that is not prefix-free Example
Huffman tree for a dyadic three-symbol source Example
Every finite binary prefix code is uniquely decodable Lemma
Binary Kraft criterion for finite positive codeword lengths Theorem
Ideal-real interval coding is prefix-free and lossless Theorem
McMillan inequality for finite binary uniquely decodable codes Theorem
Shannon bounds for finite binary codes Theorem
The deterministic Huffman code is prefix-optimal Theorem
The rational finite-register arithmetic code is prefix-free and lossless Theorem

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

Stanford EE376A, Information Theory Course Notes, Chapter 4: Lossless Compression (standard reference, not scraped)
MIT OpenCourseWare 6.450, Memory-less sources, prefix free codes, and entropy (standard reference, not scraped)