ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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.

The parabola segment $\{(x,x^2):0\leq x\leq1\}$ has content zero in $\mathbb{R}^2$

Example

The parabola segment $\{(x,x^2):0\le x\le1\}$ has content zero in $\mathbb R^2$ .

Facts & Assumptions

Given: $f(x)=x^2$ on $[0,1]$ .

[L1]

Polynomials are continuous (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, Integer powers $a^m$ ).

[L2]

Graphs of continuous functions on closed rectangles have content zero (The graph of a continuous function on a closed nondegenerate rectangle in $\mathbb{R}^m$ has content zero in $\mathbb{R}^{m+1}$ ).

Verification

technique · direct

1.1

Apply [L2] to the continuous polynomial in [L1].

L1L2

1.2

Directly, divide $[0,1]$ into $N\ge1$ equal subintervals. On each one, $|x^2-y^2|\le2/\iota(N)$ , so four squares of side $1/\iota(N)$ cover that graph piece. The resulting finite cover has total area at most $4/\iota(N)$ , which can be made arbitrarily small.

L1given

2.1

Both arguments establish content zero.

step 1.1step 1.2∎

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: 126 results over 22 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

J. Lebl, Basic Analysis, Jordan Measurable Sets (standard reference, not scraped)
J. Lebl, Basic Analysis, Outer Measure and Null Sets (standard reference, not scraped)
A. Cañez, multivariable calculus notes (standard reference, not scraped)

The parabola segment {(x,x2):0≤x≤1}\{(x,x^2):0\leq x\leq1\}{(x,x2):0≤x≤1} has content zero in R2\mathbb{R}^2R2