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.
Example
The endpoint-singular integral satisfies
Facts & Assumptions
Given: The integrand on .
The rational -test gives convergence at zero when (The improper -test for rational exponents).
The truncated power formula evaluates proper integrals (Truncated integrals of rational powers).
Verification
Since , convergence follows from [L1]. For , [L2] gives [L1, L2] As , , so the limit is two.
This is not a proper Riemann integral on : the displayed integrand is unbounded at zero, whereas every properly Riemann-integrable function is bounded.
Depends on
- The improper $p$-test for rational exponents
- Truncated integrals of rational powers
- Improper integrals at a finite singular endpoint
- Rational powers $a^r$ of a positive base
- Existence and uniqueness of $n$-th roots: a unique $a^{1/n} \ge 0$ with $(a^{1/n})^n = a$
- For bounded $f$ on $[a,b]$ and a partition $P$: the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$-th subinterval, and the lower and upper Darboux sums $L(f,P) = \sum_i m_i \Delta_i$ and $U(f,P) = \sum_i M_i \Delta_i$
Used by
Dependency tree · two levels
51 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
- William F. Trench, Introduction to Real Analysis, Examples 3.4.1–3 (standard reference, not scraped)