ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Reading the length and third coordinate of a coded finite sequence

Example

Consider the sequence $(2, 5, 1)$ . Its binary-entry blocks are $bin (2) = 10, bin (5) = 101, bin (1) = 1,$ so the code word from A natural-number coding of finite sequences is $1 1110 11010 1110101 101.$ Let $s$ be the natural number with that binary numeral. Then the decoded header has length $3$ , and the third coordinate block decodes to $1$ . Hence $len (s) = 3, entry (s, 2) = 1.$

Facts & Assumptions

Given: The coded sequence $s$ described above.

[L1]

The sequence-length and coordinate-reading functions for this coding are primitive recursive, by The sequence-length and coordinate-reading functions are primitive recursive.

Verification

technique · direct

1.1

The block $1110$ immediately after the initial sentinel has three leading $1$ 's before its first $0$ , so the decoded sequence length is $3$ .

given

1.2

After the first two payload blocks $11010$ and $1110101$ , the last block is $101$ , which is the self-delimiting block for the binary numeral $1$ . Therefore the third decoded coordinate is $1$ .

given

2.1

Steps 1.1 and 1.2 show exactly what the theorem [L1] predicts on this concrete input: the primitive-recursive decoders return $len (s) = 3$ and $entry (s, 2) = 1$ .

L1step 1.1step 1.2∎

Depends on

The sequence-length and coordinate-reading functions are primitive recursive

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Klaus Sutner, Coding Functions (standard reference, not scraped)