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 rationals

Definition

Every rational has a representative with positive denominator (if $b < 0$ then $(a, b) \sim (- a, - b)$ with $- b > 0$ , Every rational has a positive-denominator representative ↗). For representatives with $b > 0$ and $d > 0$ define

$[(a, b)] \leq [(c, d)] ⟺ a d \leq c b in Z .$

A rational $x$ is positive when $0 < x$ ; on such representatives, $0 < [(a, b)]$ exactly when $0 < a$ .

Remarks

Well-definedness, totality, and compatibility with the arithmetic: The rationals form a totally ordered field.
The integer order used on the right is Order on the integers.

Depends on

Dependency tree · 2 levels

Sources