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

1.1

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

given

2.1

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

step 1.1algebra

3.1

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

step 2.1∎

Used by

Nothing in the library uses this result yet.

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