LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Induced simplicial chain maps commute with boundaries

Statement

If $f : K \to L$ is simplicial, then $\partial f_{#} = f_{#} \partial$ on every simplicial chain group.

Proof

Given: A simplicial map $f : K \to L$ and an oriented simplex $[v_{0}, \dots, v_{n}]$ .

1.1given

If $f (v_{0}), \dots, f (v_{n})$ are pairwise distinct, then deleting one vertex before applying $f$ or after applying $f$ produces the same oriented face. Therefore $\partial f_{#} [v_{0}, \dots, v_{n}] = f_{#} \partial [v_{0}, \dots, v_{n}]$ term by term.

1.2given

If some image vertices repeat, then $f_{#} [v_{0}, \dots, v_{n}] = 0$ . In $f_{#} \partial [v_{0}, \dots, v_{n}]$ , the faces whose remaining image vertices still repeat map to $0$ , while the two faces obtained by deleting one of a repeated pair map to the same oriented simplex with opposite signs and cancel. Hence $f_{#} \partial [v_{0}, \dots, v_{n}] = 0 = \partial f_{#} [v_{0}, \dots, v_{n}]$ .

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 cover all simplices, so $\partial f_{#} = f_{#} \partial$ .

Depends on

Used by

Last vertex map is carried by original simplices Lemma
Simplicial homology is functorial Proposition
Contiguous simplicial maps induce the same map on simplicial homology Theorem

Dependency tree · two levels

6 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

Allen Hatcher, Algebraic Topology (standard reference, not scraped)
Vidit Nanda, Computational Algebraic Topology, Lecture 03: Homology (standard reference, not scraped)