IdeaWell

Project ideas from Hacker News discussions.

Finding a bug in Dummit and Foote's Abstract Algebra

Original Article

Hacker News Discussion

📝 Discussion Summary (Click to expand)

Theme 1 – Vacuous truth in empty‑set cases
"It's true precisely because it's vacuous. If you quantify over the empty set, anything is true." – tim‑kt

Theme 2 – Proof assistants as thinking tools
"After years of using the things, I believe not enough focus is given to high‑velocity uses of proof assistants for prototyping." – Paracompact

Theme 3 – Definition tweaks vs accepting degenerate cases
"I think a slightly better fix is to change definitions to allow g = { (1, {}) } to be regarded as a left‑inverse … to allow left‑inverses to be partial functions." – ajkjk

🚀 Project Ideas

VacuaCheck

Summary

  • Detects unused hypotheses and vacuously true goals in formal proof scripts (Lean, Coq, Isabelle).
  • Suggests stronger theorems by automatically removing unnecessary assumptions, reducing clutter and improving generality.

Details

Key Value
Target Audience Formal methods researchers, theorem‑proving enthusiasts, graduate students using Lean/Coq/Isabelle
Core Feature Static analysis that flags hypotheses never used in any proof step and checks for quantifiers over empty types, proposing simplified statements
Tech Stack Rust parser + Lean 4 plugin (or Coq/Isabelle extensions); language‑agnostic via OpenTheory interchange format
Difficulty Medium
Monetization Hobby

Notes

  • Quote from shmoil: “I asked AI to formalize an old important paper in analysis… It came back, and said: ‘I formalized it, it is all good, but the assumption that epsilons > 0 is not used anywhere. Shall we remove it, you a get a stronger result this way?’” – VacuaCheck would automate exactly this insight.
  • Potential for discussion: integration with proof‑assistant IDEs could spark debate on when vacuous truths should be kept vs. removed, and provide a concrete tool for improving proof clarity.

EmptyDomain Scratchpad

Summary

  • Interactive notebook‑style environment for rapid prototyping with a proof assistant that automatically highlights edge cases involving empty domains or codomains.
  • Offers real‑time warnings and suggestions (e.g., switching to image‑based left inverses or allowing partial functions) to avoid vacuous pitfalls.

Details

Key Value
Target Audience Mathematicians and students learning formal proofs, especially those working in algebra/category theory where empty sets cause trouble
Core Feature Real‑time analysis of the proof state detecting when premises allow empty types and proposing generalized definitions or special cases to maintain correctness
Tech Stack TypeScript/React frontend, CodeMirror editor, Lean 4 server via WASM/JSON‑RPC, custom tactics for emptiness detection
Difficulty High
Monetization Hobby

Notes

  • Paracompact noted proof assistants “can altogether replace scratch paper for fumbling around with new concepts.” This scratchpad would make that frictionless by catching the empty‑set issues that cause frustration (see the lengthy thread on left invertibility with A = {}).
  • Potential for discussion: could become a teaching aid that illustrates why definitions like “left inverse: g: B → A” need tweaks, and generate community consensus on better formulations.

LemmaLifter

Summary

  • Platform where users upload informal mathematical notes (LaTeX or plain text) and receive automated suggestions for formalization, including detection of missing non‑emptiness constraints and proposed definition tweaks (e.g., partial left inverses).
  • Community voting on definition adjustments to converge on best‑practice formalizations.

Details

Key Value
Target Audience Researchers formalizing existing literature, educators creating course material, hobbyist provers
Core Feature Upload text → structured draft with highlighted assumptions, automatic vacuity checks, and suggestions to add/remove constraints; includes a wiki of common fixes (partial functions, image‑based domains)
Tech Stack Python backend using LLMs fine‑tuned on math corpora (e.g., Minerva) plus Lean/AST verification; React frontend for collaboration and voting
Difficulty High
Monetization Revenue-ready: Subscription model ($10/user/mo) or pay‑per‑project for teams

Notes

  • The discussion contains multiple proposals (partial functions, image‑based g, adjusting definitions) – LemmaLifter would let users test these ideas formally and vote on which work best.
  • Potential for discussion: could drive community‑driven standards for handling vacuous truths, reduce duplicated effort, and generate concrete formalization projects that address the pain points voiced in the thread.

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