TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 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.

The deterministic time hierarchy theorem

Statement

For time-constructible $f, g$ satisfying $f (n) \log f (n) = o (g (n))$ , $D T I M E (f (n)) ⊊ D T I M E (g (n)) .$

Facts & Assumptions

Given: the stated constructible bounds.

Proof

technique · contradiction

1.1

The clocked diagonal construction decides a language $D$ in $D T I M E (g)$ .

givenconstruct

2.1

If a machine running in $O (f)$ decided $D$ , its code and a sufficiently long padded input would be a valid diagonal instance. The construction returns the opposite answer there, contradiction.

step 1.1assume-contradischarge-contradiction∎

Depends on

Used by

P is properly contained in EXP Corollary
Budgeting a time-diagonal language Example
False: unrestricted diagonalization respects any resource bound False statement
Hierarchy theorems do not separate P from NP Proposition

Dependency tree · two levels

6 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, Theorem 3.1 (standard reference, not scraped)