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.
A domino family where matching fails unless one illegally permutes symbols inside a tile
Statement refuted
PCP permits one to reorder the symbols inside a chosen domino before concatenating it.
Facts & Assumptions
Given: The one-domino PCP instance .
The companion false statement says PCP would allow internal symbol reordering, by FALSE: PCP permits reordering symbols inside a domino.
Counterexample
Consider the one-domino instance If internal permutation were allowed, the bottom word ba could be rearranged to ab, so one use of the domino would look like a match.
In actual PCP, only whole dominoes may be concatenated. Using the unique tile times gives top word and bottom word , whose first symbols differ. So the instance has no legal match.
Therefore this instance refutes the statement. It is exactly the concrete counterexample recorded abstractly in [L1].
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
2 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.