ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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.

The subsets of $X$ containing a fixed point $x$ form the principal ultrafilter at $x$

Example

Let $x \in X$ . The family

$U_{x} : = {A \subseteq X : x \in A}$

is the principal ultrafilter at $x$ . It is the filter generated by the singleton ${x}$ .

Facts & Assumptions

Given: A set $X$ , a point $x \in X$ , and the family $U_{x}$ displayed above.

[L1]

If $\emptyset \neq C \subseteq X$ , then $F_{C} = {A \subseteq X : C \subseteq A}$ is the filter generated by ${C}$ , and it is an ultrafilter exactly when $C$ is a singleton (For every nonempty $C \subseteq X$ , the supersets of $C$ form the filter generated by ${C}$ , and this filter is an ultrafilter exactly when $C$ is a singleton).

[F1]

A principal ultrafilter on $X$ is an ultrafilter of the form ${A \subseteq X : y \in A}$ for some $y \in X$ (Ultrafilter).

[L2]

A filter is an ultrafilter exactly when it contains exactly one of $A$ and $X ∖ A$ for every $A \subseteq X$ (Characterisation of ultrafilters: every set or its complement).

Verification

technique · direct

1.1

The set ${x}$ is a nonempty singleton, and ${x} \subseteq A$ holds exactly when $x \in A$ .

given

1.2

For every $A \subseteq X$ , exactly one of $x \in A$ and $x \in X ∖ A$ holds.

given

2.1

Applying [L1] to $C = {x}$ shows that $U_{x}$ is the filter generated by ${{x}}$ and is an ultrafilter.

step 1.1L1

3.1

Equivalently, step 1.2 verifies the complementary-pair condition of [L2] directly.

step 1.2step 2.1L2

4.1

By [F1], this ultrafilter is principal, and the displayed formula identifies it as the principal ultrafilter at $x$ .

step 2.1F1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Ultrafilter (set theory) (Wikipedia) (standard reference, not scraped)
Ultrafilter (Wikipedia) (standard reference, not scraped)

The subsets of XX containing a fixed point xx form the principal ultrafilter at xx