tomevault-io Lean Prover
Domain: Proof Assistant / Formal Verification
Overview
A skill for using Lean 4, a modern interactive theorem prover with powerful type theory, dependent types, and mathematical structures.
Capabilities
- Define mathematical structures and theories
- Prove theorems interactively
- Build certified programs
- Create mathematical libraries
- Implement custom automation
Key Concepts
- Dependent Types: Types depending on values
- Lean's Type Theory: Inductive types, structures, classes
- Tactics: Proof automation and construction
- Monads: State, IO, reader monads in proofs
- Metaprogramming: Custom proof automation
Lean Features
- Extensible syntax and semantics
- Natural number game
- Mathematical library (mathlib)
- Continuous predicates
- Homotopy type theory support
Use Cases
- Formal mathematics
- Program verification
- Compiler certification
- Language meta-theory
- Mechanized proofs
Canonical References
| Reference | Why It Matters |
|---|---|
| Avigad, Massot, "Mathematics in Lean" | Official Lean tutorial |
| Carneiro, "The Type Theory of Lean" (2019) | Lean's type theory foundation |
| mathlib documentation | Mathematical library examples |
| Lean 4 manual | Language reference |
Research Tools & Artifacts
Lean ecosystem:
| Tool | What to Learn |
|---|---|
| mathlib | Mathematics library |
| Lean 4 | Implementation |
| Lean 3 | Previous version |
| Verso | Documentation tool |
Research Frontiers
1. Metaprogramming
- Goal: Custom automation and tactics
- Approach: Lean 4 metaprogramming API
- Papers: "Metaprogramming in Lean 4" (2020)
2. Automation
- Goal: More powerful proof automation
- Approach: AI-assisted proving, auto-params
- Papers: "ProofNet" experiments
Implementation Pitfalls
| Pitfall | Real Consequence | Solution |
|---|---|---|
| Proof complexity | Large proofs | Structure with lemmas |
| Universe issues | Universe level errors | Explicit universe annotations |
| Definitional vs propositional | Wrong equality type | Use appropriate eq |
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