ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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.

One unit circle gives semicircle length pi, circumference 2 pi, diameter 2, and disc area pi

Example

For the unit circle and its enclosed unit disc, the geometric quantities calibrated by $π$ are

quantity	value	normalization giving $π$
semicircle length	$π$	the length itself
circumference	$2 π$	circumference divided by diameter
diameter	$2$	used as the denominator above
disc area	$π$	area divided by the square of the radius

Facts & Assumptions

Given: A circle and disc of radius $1$ .

[L1]

Every once-traversed unit semicircle has length $π$ (The arc length of a unit semicircle is pi).

[L2]

A circle of radius $r > 0$ has circumference $2 π r$ , diameter $2 r$ , and circumference-to-diameter ratio $π$ (Every circle has circumference 2 pi r and circumference-to-diameter ratio pi).

[L3]

A disc of radius $r > 0$ has area $π r^{2}$ (A disc of radius r has Riemann area pi r squared; in particular the unit disc has area pi).

Verification

technique · direct

1.1

At $r = 1$ , [L1] gives semicircle length $π$ .

givenL1

1.2

At $r = 1$ , [L2] gives circumference $2 π$ and diameter $2$ , so their ratio is $π$ .

givenL2algebra

1.3

At $r = 1$ , [L3] gives disc area $π$ .

givenL3algebra

2.1

These three substitutions give every entry and normalization in the table.

step 1.1step 1.2step 1.3∎

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

J. Lebl, Basic Analysis II, section 11.4.3 (standard reference, not scraped)