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.

Geometric simplices intersect in the realization of their common face

Statement

If $σ, τ \in K$ , then the corresponding geometric simplices in $∣ K ∣$ satisfy $∣ σ ∣ \cap ∣ τ ∣ = ∣ σ \cap τ ∣ .$

Proof

Given: Two simplices $σ, τ$ in an abstract simplicial complex $K$ .

1.1

Let $x \in ∣ σ ∣ \cap ∣ τ ∣$ . Viewed as a barycentric-coordinate function on the full vertex set of $K$ , the support of $x$ is contained in $σ$ because $x \in ∣ σ ∣$ , and it is contained in $τ$ because $x \in ∣ τ ∣$ . Hence $supp (x) \subseteq σ \cap τ$ , so $x \in ∣ σ \cap τ ∣$ .

given

1.2

Conversely, if $x \in ∣ σ \cap τ ∣$ , then $supp (x) \subseteq σ \cap τ$ , so in particular $supp (x) \subseteq σ$ and $supp (x) \subseteq τ$ . Therefore $x \in ∣ σ ∣ \cap ∣ τ ∣$ .

given

2.1

Steps 1.1 and 1.2 prove the two inclusions, so the two sets are equal.

step 1.1step 1.2∎

Depends on

Used by

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 01: Complexes (standard reference, not scraped)