Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-01
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.

ind⁡K2(G)=2∣E(G)∣ under the induced-embedding convention

Statement

For every finite simple graph G,

ind⁡K2(G)=2∣E(G)∣.

Facts & Assumptions

Given: A finite simple graph G.

[F1]

ind⁡K2(G) counts induced embeddings of the two labelled vertices of K2 into G (The induced-embedding count ind⁡H(G), Induced embeddings and induced copies of a graph).

[L1]

The sum of vertex degrees is 2∣E(G)∣ (Handshake lemma: the sum of the vertex degrees is twice the number of edges).

Verification

technique · direct
1.1

An induced embedding of K2 is exactly an ordered adjacent pair (u,v) of vertices of G.

F1
2.1

Counting ordered adjacent pairs by their first vertex gives ∑u∈V(G)deg⁡(u).

step 1.1
3.1

By the handshake lemma this is 2∣E(G)∣, proving the formula.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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