ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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 affine group of the line

Example

Assume ${A C}_{ω}$ . The matrices

$(\begin{array}{cc} a & b \\ 0 & 1 \end{array}), a > 0,$

form a connected two-dimensional Lie group. Its Lie algebra has basis $H = E_{00}$ and $X = E_{01}$ with $[H, X] = X$ .

Facts & Assumptions

Given: Coordinates $(a, b) \in R_{> 0} \times R$ .

[F1]

The tangent Lie bracket is the value at the identity of the commutator of the corresponding left-invariant fields. Lie bracket on the tangent space of a Lie group. The Lie bracket of smooth vector fields.

[F2]

Matrix units have the standard product rule. Matrix units $E_{i j}$ and the Kronecker delta.

[F3]

Countable choice is inherited through [F1]. The Axiom of Countable Choice ( ${A C}_{ω}$ ).

Verification

technique · direct

1.1

Multiplication and inversion are $(a, b) (c, d) = (a c, a d + b)$ and $(a, b)^{- 1} = (a^{- 1}, - a^{- 1} b)$ . These are smooth for $a, c > 0$ . The chart $(a, b) \mapsto (\log a, b)$ identifies the underlying manifold with $R^{2}$ , hence it is connected.

F1algebra

2.1

Tangent matrices at the identity are $u E_{00} + v E_{01}$ . In the $(a, b)$ coordinates, the left-invariant fields generated by $H$ and $X$ are $H^{L} = a \partial_{a}$ and $X^{L} = a \partial_{b}$ . Their commutator is $[H^{L}, X^{L}] = a \partial_{b} = X^{L}$ , so [F1] gives $[H, X] = X$ .

F1step 1.1algebra

3.1

The group is nonempty and two-dimensional; its nonabelian bracket has the one-dimensional ideal spanned by $X$ . No metric, nondegeneracy, interval, endpoint, or biconditional occurs. ${A C}_{ω}$ is inherited only through [F1], with no further choice.

F1F2F3step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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. (standard reference, not scraped)
Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)