Alphabeta MathHow 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.
Every prime bull-free graph is basic
Statement
Every prime bull-free graph is basic.
Facts & Assumptions
Given: A prime bull-free graph .
A bull-free graph is basic exactly when it is not composite (Basic and composite bull-free graphs).
A prime graph has no nontrivial module (Prime graphs: those whose only modules are the trivial ones).
Every composite bull-free graph has a nontrivial module (Every composite bull-free graph has a nontrivial module).
Proof
If were composite, then the composite-case theorem [L1] would supply a nontrivial module of . This contradicts [F2], so is not composite.
Since is bull-free and not composite, [F1] says that is basic.
Depends on
Used by
Dependency tree · two levels
11 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
- Maria Chudnovsky, The structure of bull-free graphs III: global structure, Theorem 4.2 (standard reference, not scraped)