ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Dependent choices as extending partial tuples

Example

For nonempty $(X_{n})_{n < ω}$ , the extension relation on finite choice tuples produces a complete choice function from a path starting at the empty tuple. For a serial relation $R$ and prescribed $a$ , finite $R$ -paths starting at $a$ recover that prescribed start even if the path-of-paths starts later.

Facts & Assumptions

[F1]

AC implies DC implies countable choice: DC on finite partial tuples yields countable choice.

[F2]

Recovering a prescribed starting point in DC: Nested finite paths starting at a prescribed point recover prescribed-start DC.

Verification

Given: The objects and hypotheses in the statement.

1.1

The first two tuple extensions are $\emptyset$ , $(x_{0})$ , $(x_{0}, x_{1})$ with $x_{0} \in X_{0}$ and $x_{1} \in X_{1}$ . In a path with one-term extensions, the $n$ th tuple has length $n$ ; its union therefore assigns exactly one permissible value at each index in omega. This is the tuple construction in the cited implication.

2.1

For finite $R$ -paths the shortest object is $(a)$ . A nested path beginning at a longer object of length $m \geq 1$ has lengths $m + n$ ; their union still has domain omega and starts at $a$ . Every adjacent pair is certified within a finite path.

F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Jech, §2.4 dependent-choice proposition, pp.22–23 (standard reference, not scraped)