Prove2Me

Adjacent vertices have one common neighbor

Proved
Conway99Formal.SrgCore.neighborhood_internal_degree

by harry · Oct 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

common-neighborsconway99-formal-project-20261003strongly-regular-graphs

In an SRG(99,14,1,2), adjacent vertices u and v have exactly one common neighbor.

∣N(u)∩N(v)∣=1(uv∈E(G))|N(u)\cap N(v)|=1\qquad(uv\in E(G))

Role: This is the adjacent common-neighbor condition in finite-set form.

Preamble
import Mathlib

namespace Conway99Formal.SrgCore
end Conway99Formal.SrgCore

set_option autoImplicit false

/-! Graph-owned parameter and adjacency identities for a hypothetical SRG(99,14,1,2).
Sources: `Conway99/Conway99/Core.lean` §§1–3, 8.1;
`Conway99/Conway99/Claims/C01srgcorealgebra.lean` §§0, 3, 6;
`Conway99/results/R005_star_complement_square_discriminant.md`.
-/

open Conway99Formal.SrgCore

open SimpleGraph Matrix Finset

variable {V : Type*} [Fintype V] [DecidableEq V]
variable (G : SimpleGraph V) [DecidableRel G.Adj]
Formal statement
theorem Conway99Formal.SrgCore.neighborhood_internal_degree (h : G.IsSRGWith 99 14 1 2)
    (u v : V) (huv : G.Adj u v) :
    (G.neighborFinset u ∩ G.neighborFinset v).card = 1 := by sorry
Source
Exact original Lean source: formalization/2026-10-03/srg-core/Core.lean#L32-L45; source commit a45708acebe3f397faccb1b646be906f24f23ee5; source SHA-256 64ce9b86d07bbd11a61266b80c3043c34c08d7f939471fff2c44dc34ff37904. Mechanically extracted declaration: blob/a45708acebe3f397faccb1b646be906f24f23ee5/formalization/2026-10-03/srg-core/Core.lean#L32-L45.

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