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.
Sharp and classical fundamental theorems of calculus agree
The first fundamental theorem: if is integrable on and continuous at , then ; in particular a continuous has as a primitive concerns a continuous integrand and its classical primitive; The second fundamental theorem: if is differentiable on with and is integrable, then assumes the differentiability hypotheses it states. The sharp theorem Fundamental theorem of calculus for absolutely continuous functions instead characterises exactly the absolutely continuous functions by an almost-everywhere derivative in and an every- reconstruction formula. Thus the classical statements are special cases, not replacements for its hypotheses.
Depends on
- Fundamental theorem of calculus for absolutely continuous functions
- The first fundamental theorem: if $f$ is integrable on $[a,b]$ and continuous at $c$, then $F'(c) = f(c)$; in particular a continuous $f$ has $F$ as a primitive
- The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G' = f$ and $f$ is integrable, then $\int_a^b f = G(b)-G(a)$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
35 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
- Christopher Heil, Absolute Continuity and the Banach--Zaretsky Theorem, §§3.1 and 3.6 (standard reference, not scraped)