PropositionStatement: 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.

Sharp-P is closed under sum and product

Statement

If $f, g \in # P$ , then $f + g$ and $f g$ belong to $# P$ .

Facts & Assumptions

Given: polynomial-time nondeterministic machines $M_{f}, M_{g}$ whose accepting-path counts are $f, g$ .

[L1]

#P functions are accepting-path counts. by Sharp-P and Gap-P functions.

Proof

technique · direct

1.1

For the sum, make one initial binary choice. On its $0$ branch simulate $M_{f} (x)$ and on its $1$ branch simulate $M_{g} (x)$ . The accepting paths are a disjoint tagged union, so their number is $f (x) + g (x)$ .

L1givenconstruct

1.2

For the product, simulate $M_{f} (x)$ ; after each accepting path, independently simulate $M_{g} (x)$ , while every rejecting $M_{f}$ path rejects. Each of the $f (x)$ accepting first-stage paths has exactly $g (x)$ accepting continuations, so the total is $f (x) g (x)$ . This also covers either count being zero.

L1givenconstructalgebra

2.1

Both composite machines run in polynomial time, so [L1] puts their two counting functions in $# P$ .

L1step 1.1step 1.2∎

Depends on

Sharp-P and Gap-P functions

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

1 result within one dependency step 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)