HOL4 vs Lean
HOL4
hol-theorem-prover.org
Lean
lean-lang.orgSide by side
| Figure | HOL4 | Lean |
|---|---|---|
| Price from | Free | Free |
| Free plan | Yes | Yes |
| Free trial | — | — |
| Open source | — | — |
| Web | — | Yes |
| Windows | Yes | Yes |
| macOS | Yes | Yes |
| Linux | Yes | Yes |
| Self-hosted | Yes | — |
| Plans published | 1 | 1 |
| Spec score · Formal Verification Tools | 8.1 (#9) | 8.6 (#2) |
| Automation | Built-in decision procedures and theorem provers can automatically establish many simple theorems. | The site highlights the Grind tactic for handling pattern matching and case analysis and solving systems of linear inequalities. |
| License | HOL is free software released under the Modified (3-clause) BSD licence. | Lean is released under the Apache 2.0 license, which the documentation describes as allowing others to use and extend the code and mathematical libraries freely. |
| Support | The community offers help through a mailing list and Zulip chat, while bug reports and feature suggestions go to GitHub issues. | The Lean Community Zulip is described as the central hub for questions and collaboration, and Lean FRO hosts weekly technical office hours. |
Plans
In their own words
HOL4
HOL4 is a free, interactive proof assistant for higher-order logic, with a programming environment for proving theorems and building proof tools. It supports work that combines deduction, execution, and property checking. Built-in decision procedures and theorem provers can establish many simple theorems, and an oracle mechanism can connect to external programs such as SMT and BDD engines. HOL4 supports higher-order logic and Standard ML, and can produce proof artifacts and counterexamples. It is self-hosted and listed for Linux, macOS, and Windows. A Standard ML compiler is required, with Poly/ML recommended; Moscow ML and MLTon are also supported for building tool executables. Windows installation can use Cygwin or the Windows Linux subsystem with Poly/ML. Moscow ML is another option, but the Windows guide notes that it is slower, lacks some Poly/ML libraries, and cannot run concurrent Holmake builds. HOL includes Emacs modes for syntax display and session interaction, and the installation guide links to Vim plugin documentation. Online materials include a tutorial, quick reference, FAQ, manuals, and generated indexes.
Lean
Lean is an open-source programming language and proof assistant for formalizing mathematics, verifying software, and writing general code. Its minimal trusted kernel checks proofs, and the project describes it as supporting correctness in mathematical, software, and hardware verification. Lean includes the Grind tool for automating some pattern matching, case analysis, and systems of linear inequalities; metaprogramming lets users add notation and proof automation. Projects are managed with Lake, the package manager bundled with Lean. Mathlib is a community-driven library with over a million lines spanning areas including algebra, analysis, topology, probability, and computer science. The recommended setup uses VS Code with the Lean 4 extension for syntax highlighting and code completion, while Neovim with lean.nvim is also well-supported. Lean runs on Linux, macOS, and Windows, and Lean4Web lets users run code in a browser. The software is free under the Apache 2.0 license. Learning resources include textbooks, tutorials, interactive games, documentation, and an API reference.