False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-02

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.

FALSE: NP means not polynomial-time solvable

Statement

NP means "not polynomial-time solvable."

Facts & Assumptions

Given: The classes $P$ and $N P$ .

[L1]

Every language in $P$ also lies in $N P$ , by $P \subseteq N P \cap c o N P$ .

[L2]

NP is a language class defined by polynomially checkable certificates, by The class NP via polynomial-time verifiers.

Refutation

technique · direct

1.1

By [L1], any language already known to be in $P$ is automatically also in $N P$ .

L1given

2.1

Step 1.1 contradicts the slogan in the statement: a polynomial-time solvable language can still belong to $N P$ . By [L2], NP is about efficiently verifiable yes-certificates, not about being known outside polynomial time.

L2step 1.1∎

Depends on

Used by

A regular parity language refutes 'NP means not polynomial' Counterexample

Dependency tree · two levels

5 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

Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)