Alphabeta Math
LemmaStatement: AI-adaptedProof: 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.

G[W]‾=G‾[W] for every vertex set W

Statement

For every finite graph G and every W⊆V(G),

G[W]‾=G‾[W]

as graphs on vertex set W.

Facts & Assumptions

Given: A graph G and W⊆V(G).

[F1]

G[W] retains exactly the edges of G with both endpoints in W (Subgraphs, induced subgraphs and spanning subgraphs).

[F2]

Complementation replaces adjacency by nonadjacency between distinct vertices (Graph isomorphisms, automorphisms and graph complements).

Proof

technique · direct
1.1

Both displayed graphs have vertex set W.

F1F2
1.2

For distinct x,y∈W, xy is an edge of G[W]‾ if and only if it is not an edge of G[W], if and only if it is not an edge of G, if and only if it is an edge of G‾[W].

F1F2
2.1

Their vertex and edge sets are equal, so the graphs are equal.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

4 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