ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-03

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.

Reducing the word $x x^{- 1} y y^{- 1} x$ leaves the reduced word $x$

Example

For distinct letters $x, y$ in an alphabet $X$ , the word

$x x^{- 1} y y^{- 1} x$

is freely equivalent to the reduced word $x$ .

Facts & Assumptions

Given: Distinct letters $x, y$ in an alphabet $X$ .

[L1]

An elementary cancellation deletes adjacent pairs $z z^{- 1}$ and $z^{- 1} z$ , and a word is reduced exactly when no such pair occurs (Words in an alphabet with formal inverses, elementary cancellation, and reduced words).

[L2]

Freely equivalent words are connected by finitely many elementary cancellations and reverse insertions (Words in an alphabet with formal inverses, elementary cancellation, and reduced words).

Verification

technique · direct

1.1

Delete the initial adjacent pair $x x^{- 1}$ to obtain $y y^{- 1} x$ .

L1given

2.1

Delete the initial adjacent pair $y y^{- 1}$ in the resulting word to obtain $x$ .

step 1.1L1

3.1

The one-letter word $x$ has no adjacent pair and is reduced, while steps 1.1 and 2.1 are a finite cancellation sequence; hence the two words are freely equivalent.

step 1.1step 2.1L1L2∎

Depends on

Words in an alphabet with formal inverses, elementary cancellation, and reduced words

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

M. Brittenham, Group Presentations, Class Notes (standard reference, not scraped)

Reducing the word xx−1yy−1xxx−1yy−1x leaves the reduced word xx