GoKawiilRocq formalization exposes edge-case flaw in Dummit and Foote's Abstract Algebra
While formalizing Dummit and Foote's algebra textbook in the Rocq proof assistant, a Recurse Center participant found that the book's first proof exercise—claiming a function is injective if and only if it has a left inverse—fails when the domain is empty and the codomain is not. In that case the empty function is vacuously injective but has no left inverse, since no function can map from a nonempty codomain to an empty domain. The author later confirmed the issue is already listed in the book's published errata.
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The case shows how formal proof tools can surface subtle logical gaps that are easy to overlook when working informally on paper, since empty-set edge cases rarely occur to a human reader. It's a small but concrete example of formal verification catching an error in a widely used, heavily vetted mathematics textbook.
- Dummit and Foote's textbook has a known errata entry acknowledging the flawed proposition.
- The counterexample uses an empty domain set to show injectivity doesn't guarantee a left inverse.
- Using the Rocq proof assistant helped surface a logical gap that's easy to miss in manual proofs.
Dummit and Foote Abstract Algebra, 3rd Edition — This is the very textbook at the center of the story — the classic Dummit and Foote "Abstract Algebra," a staple for groups, rings, and fields. Having your own copy lets you work through Proposition 1 (and hunt for edge cases like the empty-set counterexample) yourself.
See Dummit and Foote Abstract Algebra, 3rd Edition on Amazon → Affiliate link — we may earn a commission on purchases, at no extra cost to you. Product picked by AI based on this article; it is not a tested recommendation.Source: kallus.org, 2026-09-04
Published there as: “Finding a bug in Dummit and Foote's Abstract Algebra”
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