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.
The fine-cover hypothesis in the Vitali covering theorem is load-bearing
Statement refuted
Every interval cover of a bounded set admits a countable disjoint subfamily that covers the set up to a null remainder.
Facts & Assumptions
Given: The statement above.
We use the non-fine cover by left and right nested intervals.
Counterexample
Consider the family . It covers , but it is not a fine cover at any interior point.
Exactly as in FALSE: the Vitali covering theorem holds for arbitrary covers, any disjoint subfamily of has at most one left interval and at most one right interval, hence leaves a nonempty open gap. So this cover has no disjoint subfamily with null uncovered remainder.
Therefore the fine-cover hypothesis is genuinely necessary.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
7 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.