ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generated

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.

Computing the canonical order at the first levels

Example

The canonical $<_{L}$ puts $\emptyset$ first and ${\emptyset}$ second. In $L_{3}$ , both precede the two new sets ${{\emptyset}}$ and ${\emptyset, {\emptyset}}$ . Those two are compared by their least definition codes over $L_{2}$ , under the fixed formula/arity enumeration.

Facts & Assumptions

Given: ZF. Explicit differences of the first levels identify the first two elements and the two new L_3 elements; the calculation preserves dependence on the fixed code enumeration.

[F1]

The canonical definable global well-order of L: The successor construction retains the old order before new sets, then compares their least definition codes.

[F2]

The first constructible levels: The explicit L_1, L_2 and four-element L_3 calculations identify the newly appearing sets.

Verification

1.1

At L_1 the only element is empty, so it is first. The difference $L_{2} ∖ L_{1}$ is ${{\emptyset}}$ ; its only element is placed after the old empty set. Thus the first two elements are exactly as asserted.

F1F2

2.1

Subtracting the two old elements from the four-element L_3 of F2 leaves $u = {{\emptyset}}$ and $v = {\emptyset, {\emptyset}}$ . Over L_2, u is defined by $x = {\emptyset}$ using that parameter, while v is defined by $x = x$ without parameters. Their least codes need not be these displayed witnesses. F1 puts u before v exactly when its least code precedes the least code of v, and puts both after the two old elements. The answer beyond the first two therefore retains the specified coding convention.

F1F2step 1.1∎

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

Geschke Theorem 5.9 proof pp17–18 (standard reference, not scraped)