PropositionStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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.

Member equivalence is reflexive and symmetric

Statement

For members of one object, the relation $\equiv$ is reflexive and symmetric.

Facts & Assumptions

Given: Members $x : X \to A$ and $y : Y \to A$ .

[L1]

By definition, $x \equiv y$ means there are epimorphisms $u : W \to X$ and $v : W \to Y$ with $x u = y v$ (Equivalence of members).

Proof

technique · direct

1.1

Reflexivity uses $W = X$ with $u = v = 1_{X}$ . The identity is epic and $x 1_{X} = x 1_{X}$ , so $x \equiv x$ by [L1].

L1algebra

1.2

If $x \equiv y$ , choose $W, u, v$ as in [L1] with $x u = y v$ . Reading the same equality backwards gives $y v = x u$ , so $y \equiv x$ .

2.1

Therefore the member relation is reflexive and symmetric.

step 1.1step 1.2∎

Depends on

Equivalence of members

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

Saunders Mac Lane, Categories for the Working Mathematician, VIII.4 (standard reference, not scraped)