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.

The integers have no zero divisors; multiplicative cancellation

Statement

If $x, y \in Z$ are nonzero then $x y \neq 0$ . Consequently, if $x z = y z$ and $z \neq 0$ , then $x = y$ .

Facts & Assumptions

Given: Nonzero integers $x, y$ .

[L1]

$Z$ is a totally ordered ring; in particular trichotomy holds (each integer is negative, zero, or positive) and positives are closed under multiplication (The integers form a totally ordered ring).

[L2]

Ring arithmetic in $Z$ : $(- u) v = - (u v)$ , $- (- u) = u$ , and distributivity (The integers form a commutative ring).

Proof

technique · cases

1.1L1cases-exhaustive

By trichotomy, each of $x$ and $y$ is positive or negative; split on the sign pattern.

1.2assume-case bothL1

Case both positive: $0 < x y$ by closure of positives, so $x y \neq 0$ .

1.3assume-case mixedL1L2

Case exactly one negative, say $y < 0$ (the other case is identical by commutativity): $0 < - y$ , so $0 < x (- y) = - (x y)$ , hence $x y < 0$ and $x y \neq 0$ .

1.4assume-case negnegL1L2

Case both negative: $0 < - x$ and $0 < - y$ , so $0 < (- x) (- y) = x y$ , and $x y \neq 0$ .

2.1step 1.2step 1.3step 1.4cases

In every case $x y \neq 0$ : the product of nonzero integers is nonzero.

3.1step 2.1L2∎

Cancellation: if $x z = y z$ with $z \neq 0$ , then $(x - y) z = x z - y z = 0$ ; were $x - y \neq 0$ , the product would be nonzero, so $x - y = 0$ , i.e. $x = y$ .

The integers have no zero divisors; multiplicative cancellation

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources