Prove2Me

Induced edge count from internal degrees

Proved
Conway99Formal.SrgCore.half_internal_degree_sum_eq_induced_edges

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

conway99-formal-project-20261003finite-graphshandshakinginduced-subgraph

For any finite vertex subset S in a finite simple graph, half the sum of the internal degrees equals the number of edges in the induced subgraph

12∑u∈S∣N(u)∩S∣=∣E(G[S])∣\frac{1}{2}\sum_{u\in S}|N(u)\cap S|=|E(G[S])|

Role: Each induced edge contributes once at each endpoint.

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.half_internal_degree_sum_eq_induced_edges (S : Finset V) :
    (∑ u ∈ S, (S ∩ G.neighborFinset u).card) / 2 =
      (G.induce (S : Set V)).edgeFinset.card := by sorry
Source
Exact original Lean source: formalization/2026-10-03/srg-core/Core.lean#L153-L157; source commit a45708acebe3f397faccb1b646be906f24f23ee5; source SHA-256 64ce9b86d07bbdd11a61266b80c3043c34c08d7f939471fff2c44dc34ff37904. Mechanically extracted declaration: blob/a45708acebe3f397faccb1b646be906f24f23ee5/formalization/2026-10-03/srg-core/Core.lean#L153-L157.

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