How 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.
has a convergent improper integral across an interior singularity
Example
If , then where the integral is improper at the interior point .
Facts & Assumptions
Given: Reals .
A mixed integral at requires separate convergence on and (Improper integrals with several singular ends).
The exponent gives convergence at a finite endpoint (The improper -test for rational exponents).
The truncated power formula gives (Truncated integrals of rational powers).
Verification
On the left use ; on the right use . The two one-sided integrals become respectively and .
Both converge separately by [L2], as [L1] requires. Evaluating them with [L3] and adding gives the displayed value.
Depends on
- Improper integrals with several singular ends
- The improper $p$-test for rational exponents
- Truncated integrals of rational powers
- Substitution: if $\varphi$ is differentiable on $[c,d]$ with $\varphi'$ integrable and $f$ is continuous on an interval containing $\varphi([c,d])$, then $\int_{\varphi(c)}^{\varphi(d)} f = \int_c^d (f\circ\varphi)\,\varphi'$
- Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and $\int_a^b(\lambda f+\mu g) = \lambda\int_a^b f + \mu\int_a^b g$
- Existence and uniqueness of $n$-th roots: a unique $a^{1/n} \ge 0$ with $(a^{1/n})^n = a$
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 126 results over 26 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- William F. Trench, Introduction to Real Analysis, Section 3.4 (standard reference, not scraped)