LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-24

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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.
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The integer-defining relation is an equivalence relation

Statement

The relation $(a, b) \sim (c, d) ⟺ a + d = b + c$ on $N \times N$ (The integers as equivalence classes of pairs of naturals) is an equivalence relation.

Facts & Assumptions

Given: The relation $(a, b) \sim (c, d) ⟺ a + d = b + c$ on $N \times N$ .

[A1]

Addition on $N$ is commutative and associative.

[A2]

Cancellation in $N$ : if $x + z = y + z$ then $x = y$ .

Proof

technique · direct

1.1

Reflexivity: for any $(a, b)$ we have $a + b = b + a$ , so $(a, b) \sim (a, b)$ .

1.2

Symmetry: suppose $(a, b) \sim (c, d)$ , i.e. $a + d = b + c$ . Then $c + b = d + a$ , which is the defining equation for $(c, d) \sim (a, b)$ .

1.3

Suppose $(a, b) \sim (c, d)$ and $(c, d) \sim (e, f)$ , i.e. $a + d = b + c$ and $c + f = d + e$ .

given

2.1

Adding the two equations: $(a + d) + (c + f) = (b + c) + (d + e)$ .

step 1.3algebra

3.1

Regrouping both sides: $(a + f) + (c + d) = (b + e) + (c + d)$ .

step 2.1A1

4.1

Cancelling $(c + d)$ : $a + f = b + e$ , so $(a, b) \sim (e, f)$ ; the relation is transitive.

step 3.1A2

5.1

The relation is reflexive, symmetric, and transitive, hence an equivalence relation.

step 1.1step 1.2step 4.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by The integers as equivalence classes of pairs of naturals.

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)