ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passverified 2026-08-03 (gpt-5.6-sol-codex-subscription)

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.

For $n\ge1$ , the cosets of $n\mathbb{Z}$ are the $n$ congruence classes modulo $n$

Example

Let $n\ge1$ be an integer. In the additive group $(\mathbb Z,+)$ , the subgroup $n\mathbb Z$ has left cosets

$a+n\mathbb Z=\{x\in\mathbb Z:x\equiv a\pmod n\}.$

There are exactly $n$ such cosets, represented uniquely by $0,1,\ldots,n-1$ , and $[\mathbb Z:n\mathbb Z]=n$ .

Facts & Assumptions

Given: A positive integer $n$ and the additive group $\mathbb Z$ .

[L1]

The set $n\mathbb Z=\{nk:k\in\mathbb Z\}$ is a subgroup of $(\mathbb Z,+)$ (Every subgroup of $(\mathbb{Z}, +)$ is $\langle n \rangle = n\mathbb{Z}$ for exactly one natural number $n$ , The integers form a commutative ring).

[F1]

In additive notation, the coset represented by $a$ is $a+n\mathbb Z$ (Left and right cosets $gH$ and $Hg$ of a subgroup).

[F2]

The congruence $x\equiv a\pmod n$ means $n\mid(x-a)$ (Congruence modulo an integer: $a\equiv b\pmod n$ when $n\mid(a-b)$ , including the moduli $0$ and $1$ ).

[L2]

Every class modulo positive $n$ has exactly one representative $r$ with $0\le r<n$ , and there are $n$ classes (For $n\ge 1$ , every class in $\mathbb{Z}/n$ has one representative $r$ with $0\le r<n$ , so $\lvert\mathbb{Z}/n\rvert=n$ ; while $\mathbb{Z}/0$ is in bijection with $\mathbb{Z}$ ).

[F3]

The index is the finite cardinality of the coset set (The coset set $G/H$ and the index $[G:H]$ of a subgroup, The left cosets of a subgroup partition the group).

Verification

technique · direct

1.1

By [L1], $n\mathbb Z$ is a subgroup of $(\mathbb Z,+)$ . For $x\in\mathbb Z$ , one has $x\in a+n\mathbb Z$ exactly when $x=a+nk$ for some integer $k$ , equivalently $n\mid(x-a)$ , equivalently $x\equiv a\pmod n$ .

L1F1F2

2.1

By [L2], every such class has a unique representative in $\{0,\ldots,n-1\}$ , and distinct representatives give distinct cosets by step 1.1.

step 1.1L2

3.1

Thus the coset set has cardinality $n$ , so $[\mathbb Z:n\mathbb Z]=n$ .

step 2.1F3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 81 results over 24 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

UCL lecture notes, Cosets and Lagrange's theorem (standard reference, not scraped)
Thomas W. Judson, Abstract Algebra: Theory and Applications, §6.1: Cosets (standard reference, not scraped)

For n≥1n\ge1n≥1, the cosets of nZn\mathbb{Z}nZ are the nnn congruence classes modulo nnn