LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (openai/gpt-5.4)audited 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.

Rational arithmetic is well defined

Statement

Sum, product, and negation of Arithmetic on the rationals are independent of the chosen representatives.

Facts & Assumptions

Given: Pairs with $(a, b) \sim (a^{'}, b^{'})$ and $(c, d) \sim (c^{'}, d^{'})$ (The rationals as equivalence classes of pairs of integers), all second components nonzero.

[L1]

$Z$ is a commutative ring (The integers form a commutative ring).

Proof

technique · direct

1.1

By hypothesis $a b^{'} = a^{'} b$ and $c d^{'} = c^{'} d$ .

given

2.1

Product: $(a c) (b^{'} d^{'}) = (a b^{'}) (c d^{'}) = (a^{'} b) (c^{'} d) = (a^{'} c^{'}) (b d)$ , so $(a c, b d) \sim (a^{'} c^{'}, b^{'} d^{'})$ .

step 1.1L1

2.2

Sum: $(a d + c b) (b^{'} d^{'}) = (a b^{'}) (d d^{'}) + (c d^{'}) (b b^{'}) = (a^{'} b) (d d^{'}) + (c^{'} d) (b b^{'}) = (a^{'} d^{'} + c^{'} b^{'}) (b d)$ , so $(a d + c b, b d) \sim (a^{'} d^{'} + c^{'} b^{'}, b^{'} d^{'})$ .

step 1.1L1

2.3

Negation: from $a b^{'} = a^{'} b$ , $(- a) b^{'} = (- a^{'}) b$ , so $(- a, b) \sim (- a^{'}, b^{'})$ .

step 1.1L1

3.1

All three operations are independent of representatives.

step 2.1step 2.2step 2.3∎

Depends on

Used by

The rationals form a field Theorem

Cited to discharge well-definedness by Arithmetic on the rationals.

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

T. Tao, Analysis I, 3rd ed., §4.2 (standard reference, not scraped)
L. S. Krapp, Constructions of the real numbers: a set theoretical approach (Oxford, 2014) (standard reference, not scraped)