Proof of `Neighborhood-set-system representation of simple graphs`
groundedproofs/Lax683916Proofs/NeighborhoodSetSystemRepresentation.lean · lax-683916
What this proof establishes
no assumptions
Assuming the claims on the left, the claim on the right holds — checked by the archive's pipeline. Proof code is not displayed here.
Description
The neighborhood of a vertex consists exactly of its adjacent vertices.
Proof strategy
Repackage adjacency as a vertex-indexed family of sets. Symmetry and irreflexivity give the two neighborhood-system laws, and both round trips are definitionally the original relation and laws.
Attribution
Directly from the definition of the open neighborhood of a vertex.