DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-07-24

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 real numbers

Definition

The real numbers are the quotient ring

$R : = C / N$

of the ring of Cauchy sequences (Cauchy sequences form a commutative ring) by the ideal of null sequences (Null sequences form an ideal). The class of $(a_{n})$ is written $[(a_{n})]$ . Each rational $q$ maps to the class $\hat{q}$ of the constant sequence $(q, q, q, \dots)$ .

Remarks

Two Cauchy sequences define the same real exactly when their difference is a null sequence.
That $R$ is a field is The reals form a field ↗; its order is Order on the reals.

The real numbers

Definition

Remarks

Depends on

Dependency tree · 2 levels

Sources