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.

Collapsing a relation that is not transitive

Example

Take distinct nodes $a, b, c$ and $R = {(a, b), (b, c)}$ . This relation is well-founded and extensional but not transitive. Its collapse is $a \mapsto \emptyset$ , $b \mapsto {\emptyset}$ , $c \mapsto {{\emptyset}}$ .

Facts & Assumptions

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

[F1]

Every well-founded setlike extensional relation $R$ on a definable class $X$ is isomorphic to membership on a unique transitive definable class $Y$ , by a unique definable isomorphism $π : X \to Y$ . For a set domain $X$ , the isomorphism and its image are sets. This holds without ambient Foundation. (Mostowski collapse for extensional relations)

Verification

1.1

The predecessor sets are respectively $\emptyset, {a}, {b}$ , which are distinct. In any nonempty subset of the nodes, the first present node in the list $a, b, c$ has no predecessor in that subset, proving well-foundedness. Yet $a R b$ and $b R c$ hold while $a R c$ does not.

given

2.1

The collapse equation successively gives the three displayed values. Its range is transitive: the members of ${\emptyset}$ and of ${{\emptyset}}$ are already in that range. The collapse theorem makes this the unique isomorphism to a transitive membership structure.

F1step 1.1∎

Depends on

Mostowski collapse for extensional relations

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

Marks, Set Theory, Berkeley edition — 6.11 and Figure 6 p.32, reduced worked instance. (standard reference, not scraped)