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

1.1given

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

1.2given

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

2.1step 1.1step 1.2algebra

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

2.2step 1.1A1

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.

3.1step 2.1A1

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.

Integer addition and negation are well defined

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources