DefinitionDefinition: AI-adaptedProof: 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.

Order on the reals

Definition

A real $x$ is positive when it has a representative $(a_{n})$ together with a rational $δ > 0$ and an index $N$ such that $a_{n} > δ$ for all $n \geq N$ . Define

$x \leq y ⟺ x = y or y - x is positive,$

and $∣ x ∣ = x$ if $x \geq 0$ , $∣ x ∣ = - x$ otherwise.

Remarks

Independence of the representative, trichotomy, and compatibility with the field operations: The reals form a totally ordered field ↗.
The triangle inequality for this $∣ \cdot ∣$ holds because the proof of Absolute value and the triangle inequality uses only the axioms of a totally ordered field.

Order on the reals

Definition

Remarks

Depends on

Dependency tree · 2 levels

Sources