Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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 xx−1yy−1xxx^{-1}yy^{-1}x leaves the reduced word xx

Example

For distinct letters x,yx,y in an alphabet XX, the word

xx−1yy−1xxx^{-1}yy^{-1}x

is freely equivalent to the reduced word xx.

Facts & Assumptions

Given: Distinct letters x,yx,y in an alphabet XX.

[L1]

An elementary cancellation deletes adjacent pairs zz−1zz^{-1} and z−1zz^{-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 xx−1xx^{-1} to obtain yy−1xyy^{-1}x.

L1given
2.1

Delete the initial adjacent pair yy−1yy^{-1} in the resulting word to obtain xx.

step 1.1L1
3.1

The one-letter word xx 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

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: 7 results over 4 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