ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-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.

Normal functions and fixed points at omega-one

Example

The map $f : ω_{1} \to ω_{1}$ , $f (α) = ω \cdot α$ , is normal and its fixed points form a club. Iterating f from 1 gives $1, ω, ω^{2}, \dots$ , whose supremum $ω^{ω}$ is a countable fixed point. The fixed-point club has a normal increasing enumeration.

Facts & Assumptions

[F6]

Countable unions of at most countable sets, assuming ${A C}_{ω}$ : Assuming countable choice, a countable union of at most countable sets is at most countable.

[F1]

Ordinal multiplication $α \cdot β$ : Ordinal multiplication is defined by successor addition and continuity in the right argument.

[F2]

Ordinal exponentiation $α^{β}$ , with the conventions $α^{0} = 1$ and $0^{0} = 1$ : Powers are defined by right multiplication at successors and suprema at limits.

[F3]

Ordinal multiplication is associative, and $α \cdot (β + γ) = α \cdot β + α \cdot γ$ : Ordinal multiplication is associative.

[F4]

Fixed points of a normal function form a club: A normal self-map on a regular uncountable cardinal has club many fixed points.

[F5]

Clubs are ranges of normal enumerations: A club in a regular uncountable cardinal has a normal increasing enumeration.

Verification

Given: The objects and hypotheses in the statement.

1.1

For countable alpha, omega times alpha is the order type of alpha many consecutive countable blocks, hence is countable in ZFC. Multiplication by omega on the left is strictly increasing: appending one nonempty omega block strictly increases the order type, and the recursive definition is monotone in the right argument. It is continuous at nonzero limits by the same definition. Thus f is a normal self-map of omega-one.

F1F6

2.1

Associativity and induction give $f (ω^{n}) = ω^{n + 1}$ for each finite n, starting from $ω \cdot 1 = ω$ . The countable supremum of these countable ordinals is $ω^{ω} < ω_{1}$ . Continuity gives $f (ω^{ω}) = \sup_{n} ω^{n + 1} = ω^{ω}$ .

F1F2F3F6step 1.1

3.1

The normal fixed-point theorem makes the whole fixed-point set club (including zero, since $f (0) = 0$ ). The normal-enumeration theorem applies to this club and provides its normal increasing enumeration.

F4F5step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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

Welch, Lemmas 2.12–2.13 and Exercise 2.5, pp.20–21 (live original reread) (standard reference, not scraped)