DefinitionDefinition: AI-adaptedProof: AI-generatedjudge pass (z-ai/glm-5.2)audited 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 canonical net indexed by the pairs $(A, x)$ with $A$ in a filter and $x \in A$

Definition

Let $F$ be a filter on $X$ . Its derived-net index set is

$E_{F} = {(A, x) : A \in F, x \in A},$

ordered by $(A, x) ⪯ (B, y)$ when $B \subseteq A$ . It is a directed preorder: filters contain no empty set, and for two indices choose $z \in A \cap B$ , so $(A \cap B, z)$ is above both. The net derived from $F$ is

$x_{(A, x)} : = x ((A, x) \in E_{F}) .$

This construction makes no arbitrary choice, because the point $x$ is included in the index.

The canonical net indexed by the pairs $(A, x)$ with $A$ in a filter and $x \in A$

Definition

Depends on

Dependency tree · 2 levels

Sources

The canonical net indexed by the pairs (A,x)(A,x) with AA in a filter and x∈Ax∈A

Definition

Depends on

Dependency tree · 2 levels

Sources

The canonical net indexed by the pairs $(A, x)$ with $A$ in a filter and $x \in A$