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 additive and multiplicative real Lie groups

Example

Assume ${A C}_{ω}$ . The groups $(R, +)$ , $(R_{> 0}, \cdot)$ , and $(R^{\times}, \cdot)$ are one-dimensional real Lie groups. Their exponentials are respectively

$\exp_{(R, +)} (x) = x, \exp_{(R_{> 0}, \cdot)} (x) = e^{x}, \exp_{R^{\times}} (x) = e^{x} .$

The last map lands in the positive identity component.

Facts & Assumptions

Given: The displayed groups with their open-submanifold structures.

[F1]

Smooth group operations define a Lie group. Lie group.

[F2]

The Lie-group exponential is the time-one value of the invariant integral curve. Exponential map of a Lie group.

[F3]

The ordinary exponential satisfies $e^{s + t} = e^{s} e^{t}$ and has derivative $e^{t}$ . The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ . The exponential function is smooth and $(\exp)^{'} = \exp$ .

[F4]

The exponential-map interface [F2] assumes countable choice and records its use through the supplied invariant-field and completeness result. The Axiom of Countable Choice ( ${A C}_{ω}$ ).

Verification

technique · direct

1.1F1algebra

Addition and negation are smooth on $R$ ; multiplication and inversion $a \mapsto 1 / a$ are smooth on each of the open sets $R_{> 0}$ and $R^{\times}$ . Hence [F1] gives the three one-dimensional Lie groups.

2.1F2F3step 1.1

The curve $t \mapsto t x$ is the additive one-parameter subgroup with derivative $x$ at zero. The curve $t \mapsto e^{t x}$ is a multiplicative one-parameter subgroup by [F3], has derivative $x$ at zero, and stays positive. By [F2] their time-one values give the displayed formulas.

3.1F1F2F3F4step 1.1step 2.1∎

All groups are nonempty and one-dimensional; $R^{\times}$ is disconnected but the other two are connected. At $x = 0$ all exponentials give the identity. No metric, degeneracy, endpoint issue, or biconditional occurs. The assumed ${A C}_{ω}$ is used by [F2] through its stated supplier chain, with no further choice.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

25 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

Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)