ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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.

$Dih (C_{2} \times C_{2})$ is the direct product $(C_{2} \times C_{2}) \times C_{2}$

Example

The generalized dihedral group of the Klein four group is

$Dih (C_{2} \times C_{2}) ≅ (C_{2} \times C_{2}) \times C_{2} .$

Facts & Assumptions

Given: The group $A = C_{2} \times C_{2}$ .

[L1]

$Dih (A)$ is the semidirect product of $A$ by $C_{2}$ under inversion ( The generalized dihedral group $Dih (A) = A ⋊ C_{2}$ for an abelian group $A$ ).

[L2]

A semidirect product is the direct product exactly when its defining action is trivial (The canonical semidirect decomposition is an internal direct product if and only if the defining action is trivial).

Verification

technique · direct

1.1algebra

Every $a \in C_{2} \times C_{2}$ satisfies $a^{2} = 1$ , hence $a^{- 1} = a$ . The inversion automorphism is therefore the identity.

2.1step 1.1L1L2∎

The action in [L1] is trivial by step 1.1, so [L2] gives the displayed direct product.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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.

Dih⁡(C2×C2)Dih(C2​×C2​) is the direct product (C2×C2)×C2(C2​×C2​)×C2​