Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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]\overline{G[W]}=\overline G[W] for every vertex set WW

Statement

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

G[W]‾=G‾[W]\overline{G[W]}=\overline G[W]

as graphs on vertex set WW.

Facts & Assumptions

Given: A graph GG and W⊆V(G)W\subseteq V(G).

[F1]

G[W]G[W] retains exactly the edges of GG with both endpoints in WW (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 WW.

F1F2
1.2

For distinct x,y∈Wx,y\in W, xyxy is an edge of G[W]‾\overline{G[W]} if and only if it is not an edge of G[W]G[W], if and only if it is not an edge of GG, if and only if it is an edge of G‾[W]\overline 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 7 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources