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 parabola segment has content zero in
Example
The parabola segment has content zero in .
Facts & Assumptions
Given: on .
Graphs of continuous functions on closed rectangles have content zero (The graph of a continuous function on a closed nondegenerate rectangle in has content zero in ).
Verification
Apply [L2] to the continuous polynomial in [L1].
Directly, divide into equal subintervals. On each one, , so four squares of side cover that graph piece. The resulting finite cover has total area at most , which can be made arbitrarily small.
Both arguments establish content zero.
Depends on
- The graph of a continuous function on a closed nondegenerate rectangle in $\mathbb{R}^m$ has content zero in $\mathbb{R}^{m+1}$
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
- Integer powers $a^m$
- The canonical natural $\iota(n) = n \cdot 1_F$ of a field
- Canonical naturals are positive and strictly increasing
- Measure zero and content zero in $\mathbb{R}^m$ by countable and finite cube covers
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
42 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
- J. Lebl, Basic Analysis, Jordan Measurable Sets (standard reference, not scraped)
- J. Lebl, Basic Analysis, Outer Measure and Null Sets (standard reference, not scraped)
- A. Cañez, multivariable calculus notes (standard reference, not scraped)