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.

Arithmetic on the rationals

Definition

On $Q$ (The rationals as equivalence classes of pairs of integers) define, on representatives,

$[(a, b)] + [(c, d)] : = [(a d + c b, b d)], [(a, b)] \cdot [(c, d)] : = [(a c, b d)],$ $- [(a, b)] : = [(- a, b)], 0 : = [(0, 1)], 1 : = [(1, 1)],$

and, for $[(a, b)] \neq 0$ (equivalently $a \neq 0$ ), the inverse $[(a, b)]^{- 1} : = [(b, a)]$ .

Remarks

The denominators stay legal: $b d \neq 0$ because $Z$ has no zero divisors (The integers have no zero divisors; multiplicative cancellation).
Independence of representatives: Rational arithmetic is well defined ↗ for sum, product, and negation; The reciprocal on the rationals is well-defined ↗ for the reciprocal.

Depends on

Dependency tree · 2 levels

Sources