Prove2Me

Internal degree sum is even

Proved
Conway99Formal.SrgCore.internal_degree_half_sum

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

conway99-formal-project-20261003finite-graphshandshakingparity

For any finite vertex subset S, the sum of internal degrees is even; its half, rounded down by natural-number division, doubles back to the full sum

2⌊∑u∈S∣N(u)∩S∣2⌋=∑u∈S∣N(u)∩S∣2\left\lfloor\frac{\sum_{u\in S}|N(u)\cap S|}{2}\right\rfloor=\sum_{u\in S}|N(u)\cap S|

Role: This is the parity component of the handshaking identity.

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.internal_degree_half_sum (S : Finset V) :
    2 * ((∑ u ∈ S, (S ∩ G.neighborFinset u).card) / 2) =
      ∑ u ∈ S, (S ∩ G.neighborFinset u).card := by sorry
Source
Exact original Lean source: formalization/2026-10-03/srg-core/Core.lean#L147-L151; source commit a45708acebe3f397faccb1b646be906f24f23ee5; source SHA-256 64ce9b86d07bbdd11a61266b80c3043c34c08d7f939471fff2c44dc34ff37904. Mechanically extracted declaration: blob/a45708acebe3f397faccb1b646be906f24f23ee5/formalization/2026-10-03/srg-core/Core.lean#L147-L151.

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