ExampleConstruction: 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.

An even-length language sits in P, hence also in NP and coNP

Example

Let $E : = {x \in {0, 1}^{*} : ∣ x ∣ is even} .$ Then $E \in P$ , so $E \in N P \cap c o N P$ .

Facts & Assumptions

Given: The even-length language $E$ .

[L1]

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

Verification

technique · direct

1.1givenconstruct

A deterministic machine can decide $E$ by scanning the input once and toggling one parity bit in its finite control. At the end it accepts exactly when the parity is even. This runs in linear time, so $E \in P$ .

2.1L1step 1.1∎

Applying [L1] to step 1.1 gives $E \in N P \cap c o N P$ .

Depends on

$P \subseteq NP \cap coNP$

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

Michael Sipser, MIT 18.404J Theory of Computation, Lecture 14: P and NP, SAT, Poly-time Reducibility (standard reference, not scraped)