Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29
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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.

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FALSE: Tietze-equivalent finite presentations can differ on whether their word problem is solvable

Statement

Two Tietze-equivalent finite presentations can differ on whether their word problem is solvable.

Facts & Assumptions

Given: Two Tietze-equivalent finite presentations.

[L1]

Solvability of the word problem is invariant under changing between finite presentations of the same group. (Solvability of the word problem does not depend on the chosen finite generating set)

Refutation

technique · direct
1.1

Tietze-equivalent finite presentations present the same group, so [L1] applies to them.

L1given
2.1

Hence either both word problems are solvable or neither is. They cannot differ, so the statement is false.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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