ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22

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.

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The root system A_1

Example

Let $E = R α$ be a Euclidean line and $Φ = {α, - α}$ with $(α, α) = 1$ . Then $Φ$ is the reduced crystallographic root system $A_{1}$ ; its Weyl group has order $2$ , its two bases are ${α}$ and ${- α}$ , and relative to either base its Cartan matrix is the $1 \times 1$ matrix $[2]$ .

Facts & Assumptions

Given: A Euclidean line $E = R α$ with $α \neq 0$ and the set $Φ = {α, - α}$ .

[L1]

A reduced crystallographic root system is a finite spanning set of nonzero vectors closed under its root reflections with integral Cartan integers and reducedness (Reduced crystallographic Euclidean root system).

[L2]

The reflection $s_{α}$ negates $α$ , the Weyl group is the subgroup of $O (E)$ generated by the root reflections, and the diagonal entry of the Cartan matrix relative to either one-element base ${β}$ is $2 (β, β) / (β, β) = 2$ (Weyl group, Cartan matrix of a based root system).

Verification

technique · direct

1.1

$Φ$ is finite, spans $E$ , omits $0$ , and $R α \cap Φ = {\pm α}$ ; the reflection $s_{α}$ sends $α$ to $- α$ and $- α$ to $α$ , so $s_{α} (Φ) = Φ$ ; the only Cartan integers are $2 (\pm α, \pm α) / (α, α) = \pm 2$ , which are integers. Hence $Φ$ is a reduced crystallographic root system.

L1algebra

2.1

Each positive system consists of one of the two roots, so the two bases are ${α}$ and ${- α}$ . The Weyl group is ${1, s_{α}}$ of order $2$ , and relative to either one-element base the Cartan matrix is $[2]$ by [L2]. This is the rank-one system $A_{1}$ .

L2step 1.1algebra∎

Depends on

Used by

Low-rank Dynkin coincidences Example

Dependency tree · two levels

8 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

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter II (standard reference, not scraped)