ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 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.

A member chase verifying monicity

Example

Consider the inclusion $i : 2 Z ↪ Z .$ If $x : X \to 2 Z$ is a member with $i x \equiv 0$ , then $i x = 0$ on a common epic cover, so already $x \equiv 0$ . This is the member-calculus proof that $i$ is monic.

Facts & Assumptions

Given: The inclusion $i : 2 Z ↪ Z$ .

[L1]

Monicity is detected by the implication $i x \equiv 0 \Rightarrow x \equiv 0$ (Monicity is detected by members).

[L2]

Member cancellation is an equivalent reformulation (Monicity by member cancellation).

[L3]

The category $A b$ is abelian (Abelian groups form an abelian category).

Verification

technique · direct

1.1L1L3algebra

If $i x \equiv 0$ , choose an epic cover $u : W \to X$ with $i x u = 0$ . Since $i$ is the subgroup inclusion, $x u = 0$ . Hence $x \equiv 0$ .

2.1L1L2step 1.1∎

By [L1], this proves that $i$ is monic; by [L2], it is the same computation as cancellation of equal members after applying $i$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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, Theorem VIII.4.3 (standard reference, not scraped)