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.

Specializing a two-argument program to obtain a unary residual program

Example

Fix an acceptable numbering, and let $e_{+}$ be an index for the binary program that, on coded input $(u, v)$ , returns $u + v$ . Fix the first argument to be $7$ .

Facts & Assumptions

Given: An acceptable numbering, the binary addition index $e_{+}$ , and the parameter value $7$ .

[L1]

The $s$ -m- $n$ theorem supplies a total specialization map $s_{1}^{1}$ , by The s-m-n theorem.

Verification

technique · direct

1.1

By [L1], there is a total specialization map $s_{1}^{1}$ . Put $e_{7} : = s_{1}^{1} (e_{+}, 7) .$ Then for every $v$ , $φ_{e_{7}} ({Inp}_{1} (v)) ≃ φ_{e_{+}} ({Inp}_{2} (7, v)) = 7 + v .$

L1given

2.1

So $e_{7}$ is an index for the unary residual program $v \mapsto 7 + v$ . This is exactly the interpretation recorded abstractly in Effective program specialization.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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.