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.

The integers as equivalence classes of pairs of naturals

On the set $N \times N$ of pairs of natural numbers, define

$(a, b) \sim (c, d) ⟺ a + d = b + c .$

This is an equivalence relation (The integer-defining relation is an equivalence relation ↗). The integers are the quotient

$Z : = (N \times N) / \sim,$

and we write $[(a, b)]$ for the equivalence class of $(a, b)$ .

The pair $(a, b)$ encodes the formal difference $a - b$ ; the defining relation $a + d = b + c$ is the equation " $a - b = c - d$ " restated using only addition, which is all $N$ has.
$N$ and its arithmetic (commutativity, associativity, distributivity, cancellation of addition, the order) are taken as given background throughout this construction.

Depends on