TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

PP is closed under complement

Statement

If $L \in P P$ , then its complement belongs to $P P$ .

Facts & Assumptions

Given: $L \in P P$ .

[L1]

GapP values are integer differences of accepting and rejecting path counts, by Sharp-P and Gap-P functions.

[L2]

A language is in PP exactly when membership is strict positivity of a GapP function, by PP is strict sign testing for Gap-P.

Proof

technique · direct

1.1L2givenalgebra

By [L2], choose $g \in G a p P$ with $L = {x : g (x) > 0}$ . Since $g$ is integer-valued, $x \notin L ⟺ g (x) \leq 0 ⟺ 1 - g (x) > 0.$

2.1L1step 1.1construct

If $g$ is witnessed by a machine, exchanging its accepting and rejecting labels witnesses $- g$ in GapP. A deterministic accepting machine witnesses the constant-one gap, and a leading tagged branch adds two gap functions. Hence $1 - g \in G a p P$ by [L1].

3.1L2step 1.1step 2.1∎

Applying [L2] to $1 - g$ shows that $\overline{L} \in P P$ . The added $1$ is exactly what turns the non-strict condition $g \leq 0$ into strict positivity, including $g = 0$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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: A Modern Approach (standard reference, not scraped)
Lance Fortnow, Counting Complexity (standard reference, not scraped)