LemmaStatement: AI-adaptedProof: AI-adaptedprecheck 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.

Integer addition and negation are well defined

Statement

The operations $[(a, b)] + [(c, d)] = [(a + c, b + d)]$ and $- [(a, b)] = [(b, a)]$ on $Z$ (Arithmetic on the integers) are independent of the chosen representatives.

Facts & Assumptions

Given: Pairs with $(a, b) \sim (a^{'}, b^{'})$ and $(c, d) \sim (c^{'}, d^{'})$ in the sense of The integers as equivalence classes of pairs of naturals.

[A1]

Addition on $N$ is commutative and associative.

Proof

technique · direct

1.1

By hypothesis $a + b^{'} = b + a^{'}$ .

given

1.2

By hypothesis $c + d^{'} = d + c^{'}$ .

given

2.1

Adding the two equations: $(a + b^{'}) + (c + d^{'}) = (b + a^{'}) + (d + c^{'})$ .

step 1.1step 1.2algebra

2.2

Commuting the equation of step 1.1: $b + a^{'} = a + b^{'}$ , which is the defining equation for $(b, a) \sim (b^{'}, a^{'})$ ; negation is well defined.

step 1.1A1

3.1

Regrouping both sides: $(a + c) + (b^{'} + d^{'}) = (b + d) + (a^{'} + c^{'})$ , which is the defining equation for $(a + c, b + d) \sim (a^{'} + c^{'}, b^{'} + d^{'})$ ; addition is well defined.

step 2.1A1

4.1

Both operations are independent of representatives.

step 3.1step 2.2∎

Depends on

Used by

The integers form a commutative ring Theorem

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

Dependency tree · two levels

10 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., §4.1 (standard reference, not scraped)
L. S. Krapp, Constructions of the real numbers: a set theoretical approach (Oxford, 2014) (standard reference, not scraped)