PropositionStatement: Literature-sourcedProof: AI-adaptedjudge 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.

NP is contained in IP

Statement

$N P \subseteq I P$ .

Proof

Given: a language $L \in N P$ and its polynomial-time verifier.

1.1

By The class NP via polynomial-time verifiers, on input $x$ the prover may send a polynomially bounded certificate $w$ , and the verifier can test it in polynomial time.

given

2.1

If $x \in L$ , some $w$ is accepted, whereas if $x \notin L$ , no $w$ is accepted. This is a one-round protocol with completeness $1$ and soundness $0$ , hence is an IP protocol by The class IP.

step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Arora and Barak, §8.2 (standard reference, not scraped)