ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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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.

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Integration by parts for absolutely continuous functions

Example

Assume the Axioms of Countable Choice and Dependent Choice. On $[0, 1]$ , take $F (x) = \sqrt{x}$ and $G (x) = x$ . Both are absolutely continuous, and integration by parts gives $\int_{0}^{1} \sqrt{x} d x + \int_{0}^{1} \frac{x}{2 \sqrt{x}} d x = 1.$

Facts & Assumptions

Given: Countable choice, dependent choice, $F (x) = \sqrt{x}$ , and $G (x) = x$ on $[0, 1]$ .

Verification

technique · direct

1.1

$F (x) = \int_{0}^{x} (2 \sqrt{t})^{- 1} d t$ and $G (x) = \int_{0}^{x} 1 d t$ . Both integrands are in $L^{1} [0, 1]$ , so The indefinite integral of an $L^{1}$ function is absolutely continuous makes $F$ and $G$ AC; their derivatives are respectively $(2 \sqrt{x})^{- 1}$ and $1$ almost everywhere.

givenalgebra

2.1

Apply Integration by parts for absolutely continuous functions: its endpoint term is $F (1) G (1) - F (0) G (0) = 1$ .

step 1.1

3.1

The two integrands are respectively $\sqrt{x}$ and $\sqrt{x} / 2$ , whose integrals are $2 / 3$ and $1 / 3$ .

step 2.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Donald L. Cohn, Measure Theory, 2nd ed., Corollary 6.3.9 (standard reference, not scraped)