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 $[v_{0}, \dots, v_{n}]$ .

1.1

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.

given

1.2

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 $f_{#} \partial [v_{0}, \dots, v_{n}] = 0 = \partial f_{#} [v_{0}, \dots, v_{n}]$ .

given

2.1

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

step 1.1step 1.2∎

Depends on

Used by

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)