ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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 nice name for one Cohen coordinate

Statement

For $ξ < λ$ , the canonical $Add (ω, λ)$ -name whose $n$ -th antichain consists of conditions assigning value $1$ at $(ξ, n)$ is a nice name and evaluates to the set of $1$ -bits of $c_{ξ}$ .

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

Nice names for subsets of a ground-model set defines nice names.

[F2]

Cohen, collapse, and Lévy-collapse forcing orders defines the finite-function order.

[F3]

Cohen coordinates are distinct and mutually generic defines $c_{ξ}$ from the generic union.

Proof

1.1F1F2

For $n \in ω$ , let $A_{n}$ be the conditions $p$ with $p (ξ, n) = 1$ and no other coordinates in their domains. In this displayed canonical version $A_{n}$ is the singleton containing ${((ξ, n), 1)}$ , hence an antichain; equivalently one may use any maximal antichain deciding that bit and retain only its value- $1$ part. Thus ${\dot{c}}_{ξ} = {⟨ \overset{ˇ}{n}, p ⟩ : n \in ω, p \in A_{n}}$ is nice.

2.1F2F3∎

By name evaluation, $n \in ({\dot{c}}_{ξ})_{G}$ exactly when some $p \in G$ sets $(ξ, n)$ to $1$ . Directedness makes this equivalent to $(⋃ G) (ξ, n) = 1$ , which is precisely $n \in c_{ξ}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Karagila, Forcing & Symmetric Extensions, Cohen names (standard reference, not scraped)