ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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.

$|x|$ is Lipschitz and absolutely continuous but not $C^1$ on $[-1,1]$

Example

The absolute-value function separates two implications in the hierarchy: it is Lipschitz, hence absolutely continuous, but is not even differentiable at the origin.

Facts & Assumptions

Given: The function $f(x)=|x|$ on $[-1,1]$ .

[L1]

The reverse triangle inequality gives $||x|-|y||\le|x-y|$ (The reverse triangle inequality).

[L2]

Every Lipschitz function on a compact interval is absolutely continuous ( $C^1$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation).

[L3]

Differentiability requires a two-sided difference-quotient limit (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).

Verification

technique · direct

1.1

By [L1], $f$ is $1$ -Lipschitz, and [L2] makes it absolutely continuous.

L1L2

2.1

At zero, $(|h|-|0|)/h$ equals $1$ for $h>0$ and $-1$ for $h<0$ . The two one-sided limits differ, so [L3] shows that $f'(0)$ does not exist. Therefore $f$ is not $C^1$ on $[-1,1]$ .

L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 74 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Christopher Heil, Absolute Continuity and the Banach-Zaretsky Theorem, Section 2 (standard reference, not scraped)

∣x∣|x|∣x∣ is Lipschitz and absolutely continuous but not C1C^1C1 on [−1,1][-1,1][−1,1]