ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

A radius- $2$ , height- $3$ cylinder and cone have volumes $12 π$ and $4 π$

Example

A right circular cylinder and a right circular cone both have radius $2$ and height $3$ . Their volumes are respectively $12 π$ and $4 π$ , so the cone has one third of the cylinder's volume.

Facts & Assumptions

Given: Radius $R = 2$ and height $h = 3$ .

[F1]

A right circular cylinder of radius $R \geq 0$ and height $h \geq 0$ has volume $π R^{2} h$ (A right circular cylinder of radius $R$ and height $h$ has volume $π R^{2} h$ ).

[F2]

A right circular cone of radius $R \geq 0$ and height $h \geq 0$ has volume $π R^{2} h / 3$ (A right circular cone of radius $R$ and height $h$ has volume $π R^{2} h / 3$ ).

Verification

technique · direct

1.1F1algebra

By [F1], the cylinder volume is $π \cdot 2^{2} \cdot 3 = 12 π$ .

2.1F2step 1.1algebra∎

By [F2], the cone volume is $π \cdot 2^{2} \cdot 3 / 3 = 4 π = (12 π) / 3$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Sigurd Angenent, Math 221 lecture notes, Chapter 8 §3 (standard reference, not scraped)

A radius-22, height-33 cylinder and cone have volumes 12π12π and 4π4π