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.

Compiling between two acceptable numberings

Example

Let $(α_{e})$ be the fixed machine-coding numbering, and define a second acceptable numbering $(β_{j})$ by duplicating every $α$ -index. Write $π (j) : = {\begin{cases} e, & j = ⌜ b, e ⌝_{s e q} for some b \in {0, 1}, \\ 0, & otherwise, \end{cases}$ and set $β_{j} : = α_{π (j)} .$ Then $β$ carries two distinguished syntactic names $⌜ 0, e ⌝_{s e q}$ and $⌜ 1, e ⌝_{s e q}$ for each $α$ -program, while all other indices collapse to $α_{0}$ .

Facts & Assumptions

Given: The duplicate numbering $(β_{j})$ built from $(α_{e})$ above.

[L1]

Any two acceptable numberings admit total computable translators in both directions, by Acceptable numberings are computably intertranslatable.

Verification

technique · direct

1.1

The map $t_{α \to β} (e) : = ⌜ 0, e ⌝_{s e q}$ is total computable and satisfies $β_{t_{α \to β} (e)} = α_{e} .$ The decoder-projection $t_{β \to α} (j) : = π (j)$ is also total computable, and by definition of $β$ it satisfies $α_{t_{β \to α} (j)} = β_{j}$ for every index $j$ .

givenconstruct

2.1

Thus these two numberings are explicitly compiled into each other. This is a concrete instance of [L1].

L1step 1.1∎

Depends on

Acceptable numberings are computably intertranslatable

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.