DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Absolute values on a field

Let $F$ be a field. An absolute value on $F$ is a function $∣ \cdot ∣ : F \to R_{\geq 0}$ such that for all $x, y \in F$ :

$∣ x ∣ = 0 ⟺ x = 0, ∣ x y ∣ = ∣ x ∣ ∣ y ∣, ∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣ .$

It is nonarchimedean when it satisfies the stronger inequality

$∣ x + y ∣ \leq \max {∣ x ∣, ∣ y ∣}$

for all $x, y \in F$ .

It is trivial when $∣ x ∣ = 1$ for every nonzero $x \in F$ .

Used by

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