ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedjudge 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.

Parsing and evaluating a membership formula

Example

In the membership structure on $M = {0, 1}$ , where $0 = \emptyset$ and $1 = {0}$ , let $ϕ = \exists v_{1} (v_{1} \in v_{0})$ . Then $FV (ϕ) = {v_{0}}$ . It is false when $s (v_{0}) = 0$ and true when $s (v_{0}) = 1$ .

Facts & Assumptions

Given: Work in ZF unless the statement explicitly weakens or supplements it; fix the objects and hypotheses of the statement.

[F1]

Assignments agreeing on $FV (e)$ give equal denotations when $e$ is a term and the same truth value when $e$ is a formula. Also, passage to a reduct preserves denotations and satisfaction for expressions in the smaller signature. In particular sentence truth is independent of assignment; the truth of a formula can be specified by any tuple assigning all its free variables. (Coincidence for term values and satisfaction)

Verification

1.1F1

The outer code is $(e x i s t s, v_{1}, (r e l, \in, (v a r, v_{1}), (v a r, v_{0})))$ . The body has both variables free; the outer binder removes $v_{1}$ . By coincidence all other assignment values are irrelevant.

2.1step 1.1given∎

The only membership pair in $M^{2}$ is $(0, 1)$ . For value $0$ at $v_{0}$ , neither candidate witness is a member of $0$ . For value $1$ , witness $0$ works. These are the two asserted truth values.

Depends on

Coincidence for term values and satisfaction

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Weiss, An Introduction to Set Theory (2014) — chapter 1 pp.17–18, nonempty-set formula. (standard reference, not scraped)