DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Totally ordered abelian groups

A totally ordered abelian group is an abelian group $(Γ, +, 0)$ together with a total order $\leq$ such that translation preserves the order:

γ \leq δ ⟹ γ + η \leq δ + η

for all $γ, δ, η \in Γ$ .

The positive cone is

Γ_{\geq 0} : = {γ \in Γ : γ \geq 0},

and similarly $Γ_{> 0}$ , $Γ_{\leq 0}$ , and $Γ_{< 0}$ .

Used by

Nothing. This result depends on no other item in the library.