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.

FP is contained in Sharp-P

Statement

$F P \subseteq # P$ for the page's nonnegative integer-valued function convention.

Facts & Assumptions

Given: $f \in F P$ .

[L1]

An FP function is total, nonnegative, binary-valued, and computable in polynomial time, by The nonnegative function class FP.

[L2]

A $# P$ function counts accepting paths of a binary-branching polynomial-time nondeterministic machine, by Sharp-P and Gap-P functions.

Proof

technique · direct

1.1L1givenconstruct

On input $x$ , compute $m = f (x)$ using [L1] and let $k = \max (1, ∣ bin (m) ∣)$ . Make exactly $k$ binary nondeterministic choices, producing one path for every integer $u \in [0, 2^{k})$ . Accept iff $u < m$ .

2.1step 1.1algebra

Exactly the $m$ values $u = 0, \dots, m - 1$ accept; all power-of-two padding values reject. This remains correct at $m = 0$ , when no path accepts. The output length and hence $k$ are polynomially bounded by the running time in [L1].

3.1L2step 2.1∎

The constructed machine runs in polynomial time and has accepting-path count $f (x)$ , so $f \in # P$ by [L2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

2 results 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)