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 reduction for a differentiable integrator gives
Facts & Assumptions
Given: and on .
A integrator satisfies (A continuously differentiable integrator reduces Stieltjes integration to ordinary integration).
The working FTC evaluates an integral from an antiderivative (The second fundamental theorem: if is differentiable on with and is integrable, then ).
Verification
Directly from the difference quotient, [given] so . Likewise, by expanding the difference quotient.
Therefore [L1] and [L2] give [L1, L2] ∎
Depends on
- A continuously differentiable integrator reduces Stieltjes integration to ordinary integration
- The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set
- The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G' = f$ and $f$ is integrable, then $\int_a^b f = G(b)-G(a)$
- Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and $\int_a^b(\lambda f+\mu g) = \lambda\int_a^b f + \mu\int_a^b g$
- Integer powers $a^m$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
44 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
- W. Rudin, Principles of Mathematical Analysis, Ch. 6, Theorem 6.17 (standard reference, not scraped)