LemmaStatement: AI-adaptedProof: 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.

A dovetailing schedule reaches every finite simulation stage

Statement

In the stage-sum dovetailing schedule of A dovetailing schedule, every pair $(i, t) \in N^{2}$ appears at the finite global stage $i + t$ . In particular, every fixed finite simulation prefix of every fixed task is eventually reached.

Facts & Assumptions

Given: The stage-sum dovetailing schedule $S_{0}, S_{1}, S_{2}, \dots$ .

[L1]

By A dovetailing schedule, $S_{s}$ consists exactly of the pairs $(i, t)$ with $i + t = s$ .

Proof

technique · direct

1.1

Fix $(i, t) \in N^{2}$ and set $s : = i + t$ . Then the defining condition for $S_{s}$ is satisfied, so [L1] gives $(i, t) \in S_{s}$ .

L1given

2.1

Therefore the task with index $i$ is run through its local time $t$ by the finite global stage $s = i + t$ . This is exactly the claimed fairness property.

step 1.1∎

Depends on

A dovetailing schedule

Used by

Domains and ranges of partial computable functions are computably enumerable Theorem

Dependency tree · two levels

2 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

John Watrous, Introduction to the Theory of Computing, Lecture 17: More undecidable languages; reductions (standard reference, not scraped)