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.
Immediate rejection is a halting computation, not a divergence
Statement refuted
The statement "rejection is divergence" is false.
Facts & Assumptions
Given: The machine and input from FALSE: rejection is divergence: states , tape rule and input word .
The statement refuted is: rejecting an input means diverging on that input.
A configuration is rejecting exactly when its state is , by Initial, accepting, and rejecting configurations.
Counterexample
Starting from the initial configuration on input , one legal step sends the machine to a configuration with state .
By [L1], that next configuration is rejecting, so the computation halts after one step. Hence the input is rejected and does not diverge.
Therefore this machine on input is a counterexample to [A1].
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
5 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
- Richard Zach, Sets, Logic, Computation: An Open Introduction to Metalogic (standard reference, not scraped)