ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-05

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.

A worked clause-cluster instance witnessing that CLIQUE is NP-complete

Example

Consider the formula $φ = (x \lor y \lor z) \land (\neg x \lor y \lor z) \land (x \lor \neg y \lor z) .$

Form a graph with one vertex for each of the nine displayed literal occurrences. Join two vertices exactly when they come from different clauses and their literal labels are not complementary.

Facts & Assumptions

Given: The above $3$ -CNF formula $φ$ .

[F1]

A CLIQUE instance asks for $k$ pairwise adjacent vertices, by Clique, independent set, and vertex cover decision problems.

[L1]

The reduction theorem states that the clause-occurrence construction gives a CLIQUE yes-instance exactly when the input formula is satisfiable, by 3SAT polynomial-time many-one reduces to CLIQUE.

[L2]

CLIQUE is NP-complete, by CLIQUE is NP-complete.

Verification

technique · direct

1.1

In the graph defined in the example, choose the vertices labeled by the literal $z$ in the first, second, and third clauses. These occurrences come from different clauses and no two are complementary, so every pair is adjacent. Thus they form a clique of size $3$ .

F1givenconstruct

2.1

The assignment $z = 1$ satisfies all three clauses of $φ$ , so the explicitly constructed clique agrees with the iff asserted by [L1]. This concrete yes-instance therefore illustrates the reduction theorem and, through [L2], the NP-completeness conclusion for CLIQUE.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)