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

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.

Continuity, almost-everywhere differentiability, and an integrable derivative imply Newton--Leibniz

If $F$ is continuous, differentiable almost everywhere, and $F^{'} \in L^{1}$ , then $F (x) - F (a) = \int_{a}^{x} F^{'}$ for every $x$ .

Given: The standard Cantor--Lebesgue function $C : [0, 1] \to [0, 1]$ .

technique · direct

1.1given

The function $C$ is continuous, satisfies $C (0) = 0$ and $C (1) = 1$ , and has derivative $0$ almost everywhere; thus $C^{'} \in L^{1} [0, 1]$ .

2.1step 1.1algebra

Its endpoint increment is $1$ , whereas $\int_{0}^{1} 0 = 0$ .

3.1step 2.1∎

Thus the asserted every- $x$ reconstruction formula fails without an additional hypothesis such as AC.

Used by

Nothing in the library uses this result yet.

Nothing. This result depends on no other item in the library.