ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-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.

Independent and deterministic finite information

Example

For finite $X, Y$ , independence gives $I (X; Y) = 0$ . If $Y = f (X)$ for a function $f$ on the finite alphabet of $X$ , then $I (X; Y) = H (Y)$ .

Facts & Assumptions

Given: Finite $X, Y$ with joint pmf $p_{X Y}$ ; in the second case $Y = f (X)$ .

[F1]

Finite mutual information: $I (X; Y)$ is the finite KL divergence between $p_{X Y}$ and $p_{X} \otimes p_{Y}$ .

[F2]

Finite entropy and mutual-information chain rules: $I (X; Y) = H (Y) - H (Y ∣ X)$ .

[F3]

Finite conditional entropy: conditional entropy is the average of the entropies of the conditional pmfs.

[F4]

Finite distributions, entropy, divergence, and stochastic matrices: $H (r) = - \sum_{r_{i} > 0} r_{i} \ln r_{i}$ , with $0 \ln 0 = 0$ .

Verification

1.1givenF1algebra

If $X, Y$ are independent, then $p_{X Y} (x, y) = p_{X} (x) p_{Y} (y)$ at every coordinate. Each positive-support ratio in [F1] is $1$ , so every summand is zero and $I (X; Y) = 0$ .

2.1givenF2F3F4algebra∎

If $Y = f (X)$ , then for each $x$ with $p_{X} (x) > 0$ , $p_{Y ∣ X = x}$ has mass $1$ at $f (x)$ and $0$ elsewhere, so [F4] gives entropy $- 1 \ln 1 = 0$ . Averaging these conditional entropies by [F3] gives $H (Y ∣ X) = 0$ , and [F2] yields $I (X; Y) = H (Y)$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Robert M. Gray, Entropy and Information Theory, corrected first edition (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, author prepublication (standard reference, not scraped)