Alphabeta Math
PropositionStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Free equivalence is an equivalence relation and concatenation respects it

Statement

For words on X⊔X−1X\sqcup X^{-1}, free equivalence is an equivalence relation in the sense of Equivalence relation, equivalence class, and the quotient set A/∼A/{\sim}. It is also a congruence for concatenation: if w∼w′w\sim w' and v∼v′v\sim v', then wv∼w′v′wv\sim w'v'.

Facts & Assumptions

Given: A set XX and finite words u,v,w,w′,v′u,v,w,w',v' on X⊔X−1X\sqcup X^{-1}.

[F1]

Words are freely equivalent if one can be transformed into the other by finitely many elementary cancellations and their reverse insertions (Words in an alphabet with formal inverses, elementary cancellation, and reduced words).

Proof

technique · direct
1.1

The empty sequence of elementary moves carries every word ww to itself, so w∼ww\sim w.

F1
1.2

If a finite sequence of elementary moves carries ww to w′w', reversing its order and interchanging every cancellation with the corresponding insertion gives a finite sequence from w′w' to ww; hence w∼w′w\sim w' implies w′∼ww'\sim w.

F1
1.3

If w∼w′w\sim w' and w′∼w′′w'\sim w'', concatenating the two finite move sequences gives a finite move sequence from ww to w′′w''; hence free equivalence is transitive.

F1
1.4

If one elementary move changes ww to w′w', then the same adjacent pair can be deleted or inserted inside uwvuwv, so the move changes uwvuwv to uw′vuw'v; applying this to every move in a finite sequence gives w∼w′⇒uwv∼uw′vw\sim w'\Rightarrow uwv\sim uw'v.

F1
2.1

If w∼w′w\sim w' and v∼v′v\sim v', step 1.4 gives wv∼w′vwv\sim w'v and w′v∼w′v′w'v\sim w'v'; transitivity gives wv∼w′v′wv\sim w'v'. Thus steps 1.1 through 1.3 prove that ∼\sim is an equivalence relation, and this step proves the congruence claim.

step 1.1step 1.2step 1.3step 1.4∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 24 results over 12 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