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 preserving permutation on two equal atoms
Example
On , and , the swap , is a measure-preserving probability transformation.
Facts & Assumptions
A measurable self-map preserves measure exactly when every measurable inverse image has the original measure Measure-preserving transformations and systems.
Verification
Given: The objects and hypotheses in the statement.
The masses of are respectively . A disjoint countable family has at most two nonempty members, so adding their cardinalities proves countable additivity of . Every inverse image is a subset of X and is measurable.
The inverse images of those four sets are , respectively. Their masses are unchanged, so for every measurable E. This is measure preservation.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
3 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
- Einsiedler–Ward Definition 2.1, p.13; explicit finite specialization (standard reference, not scraped)