ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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 real line is $R ⋊ R^{\times}$

Example

The group of affine bijections of the real line is

$Aff (R) ≅ (R, +) ⋊ R^{\times},$

where $a \in R^{\times}$ acts by $x \mapsto a x$ .

Facts & Assumptions

Given: The additive group $N = (R, +)$ and multiplicative group $H = R^{\times}$ .

[L1]

An action by automorphisms makes $N \times H$ a semidirect-product group ( The semidirect-product multiplication makes $N \times H$ a group).

[L2]

A holomorph acts by affine permutations $x \mapsto g α (x)$ ( The holomorph acts faithfully on $G$ by affine permutations $x \mapsto g α (x)$ ).

Verification

technique · direct

1.1

Each nonzero $a$ acts on $(R, +)$ by the automorphism $x \mapsto a x$ , and multiplication of scalars composes these automorphisms. Thus [L1] gives the group law $(b, a) (d, c) = (b + a d, a c)$ .

L1algebra

2.1

Associate $(b, a)$ with $f_{b, a} (x) = a x + b$ . Composition satisfies $f_{b, a} \circ f_{d, c} (x) = a c x + (a d + b)$ , so the association is a homomorphism by step 1.1. It is bijective because an affine map uniquely determines its slope $a$ and intercept $b$ . This is the affine action described in [L2].

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 30 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Peter J. Cameron, The Holomorph of a Group (standard reference, not scraped)

The affine group of the real line is R⋊R×R⋊R×