CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-11

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.

In $⟨ a, b ∣ a b ⟩$ , the trivial word $b a$ is stuck under free cancellation and delete-only relator rewriting

Statement refuted

Consider the following specific syntactic procedure for a displayed finite presentation: at each step, freely cancel an adjacent inverse pair or delete a contiguous occurrence of one displayed relator or its inverse. Never insert a relator and never lengthen the word.

The false claim is that this delete-only procedure reduces every word that represents the identity to the empty word. In $⟨ a, b ∣ a b ⟩$ , the word $b a$ represents the identity but admits no step at all.

Facts & Assumptions

Given: The presentation $G = ⟨ a, b ∣ a b ⟩$ and the delete-only procedure just stated.

[F1]

In a presented group, every displayed relator becomes the identity (Group presentation by generators and relations).

[F2]

In a group, an equation $y x = e$ determines $y = x^{- 1}$ (Group and abelian group).

Counterexample

technique · direct

1.1

The relation $a b = e$ gives $b = a^{- 1}$ by [F2], and therefore $b a = a^{- 1} a = e$ in $G$ .

F1F2

1.2

The word $b a$ has no adjacent inverse pair and contains neither the relator $a b$ nor its inverse $b^{- 1} a^{- 1}$ as a contiguous subword.

given

2.1

Thus $b a$ represents the identity by step 1.1 but is stuck and nonempty under the stated procedure by step 1.2, refuting the claim.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

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.

In ⟨a,b∣ab⟩⟨a,b∣ab⟩, the trivial word baba is stuck under free cancellation and delete-only relator rewriting