LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Read-only work space becomes all-tapes space after charging input

Statement

For every $s$ , $D W O R K S P A C E (s (n)) \subseteq D S P A C E (n + s (n))$ under the repository's all-tapes convention.

Facts & Assumptions

Given: a read-only-input decider using $O (s (n))$ work cells.

Proof

technique · direct

1.1

Regard its endmarked input tape as an ordinary tape. By the local-model definition the input head is confined to its $n + 2$ input positions, and the machine visits $O (s (n))$ work cells.

givenconstruct

2.1

Therefore the total visited-cell count is $O (n + s (n))$ , which is the claimed all-tapes DSPACE bound.

step 1.1algebra∎

Depends on

Used by

L is properly contained in PSPACE Corollary

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

Arora and Barak, Computational Complexity, Definition 4.1 (standard reference, not scraped)