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Proof assistant

From Wikipedia, the free encyclopedia
(Redirected from Interactive theorem proving)
For verification in computer science, see Formal verification.
Not to be confused with Interactive proof system.
An interactive proof session in RocqIDE, showing the proof script on the left and the proof state on the right

In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human–machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer.

A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics.[1]

Automated proof checking

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Automated proof checking is the process of using software for checking proofs for correctness. It is one of the most developed fields in automated reasoning. Automated proof checking differs from automated theorem proving in that automated proof checking simply mechanically checks the formal workings of an existing proof, instead of trying to develop new proofs or theorems itself. Because of this, the task of automated proof verification is much simpler than that of automated theorem proving, allowing automated proof checking software to be much simpler than automated theorem proving software.

Because of this small size, some automated proof checking systems can have less than a thousand lines of core code, and are thus themselves amenable to both hand-checking and automated software verification. The Mizar system, HOL Light, and Metamath are examples of automated proof checking systems. Automated proof checking can be done either as a batch operation, or interactively, as part of an interactive theorem proving system.

History

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Automath, which was developed by Nicolaas Govert de Bruijn starting in 1967, is often considered the first proof checker and the first system to utilize the Curry–Howard correspondence between programs and proofs.[2] Automath was used by L.S. van Benthem Jutting in 1977 to formalize Landau's Foundations of Analysis, which was the first formalization of the real numbers.[3]

In 1973, Robert Boyer and J Moore published Proving Theorems about LISP Functions which aimed to verify programs, not mathematics.[4] Their theorem prover is now known as ACL2.

In the 1970s, Edinburgh LCF introduced the idea of using a functional programming language as the metalanguage for a theorem prover, and led to the HOL family of proof assistants.[3]

The 1990s saw the rise of Rocq, (then known as Coq), which has been used for many large-scale formalization projects. Since the late 2010s, Lean, a proof assistant strongly influence by Rocq, has become another popular choice, especially for formalizing mathematics.

System comparison

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See also: Dependent type § Comparison, and Automated theorem proving § Comparison
Name Latest version Developer(s) Implementation language Features
Higher-order logic Dependent types Small
kernel		 Proof automation Proof by
reflection		 Code generation
ACL2 8.3 Matt Kaufmann, J Strother Moore Common Lisp No Untyped No Yes Yes[5] Already executable
Agda 2.8.0[6] Ulf Norell, Nils Anders Danielsson, and Andreas Abel (Chalmers and Gothenburg)[6] Haskell[6] Yes
[citation needed]		 Yes
[7]		 Yes
[citation needed]		 No
[citation needed]		 Partial
[citation needed]		 Already executable
[citation needed]
Albatross 0.4 Helmut Brandl OCaml Yes No Yes Yes Unknown Not yet implemented
F* repository Microsoft Research and INRIA F* Yes Yes No Yes Yes[8] Yes
HOL Light repository John Harrison OCaml Yes No Yes Yes No No
HOL4 Kananaskis-13 (or repo) Michael Norrish, Konrad Slind, and others Standard ML Yes No Yes Yes No Yes
Idris 2 0.6.0 Edwin Brady Idris Yes Yes Yes Unknown Partial Yes
Isabelle Isabelle2025 (March 2025) Larry Paulson (Cambridge), Tobias Nipkow (München) and Makarius Wenzel Standard ML, Scala Yes No Yes Yes Yes Yes
Lean v4.28.0-rc1[9] Leonardo de Moura (AWS) C++, Lean Yes Yes Yes Yes Yes Yes
LEGO 1.3.1 Randy Pollack (Edinburgh) Standard ML Yes Yes Yes No No No
Metamath v0.198[10] Norman Megill ANSI C
Mizar 8.1.11 Białystok University Free Pascal Partial Yes No No No No
Nqthm
NuPRL 5 Cornell University Common Lisp Yes Yes Yes Yes Unknown Yes
PVS 6.0 SRI International Common Lisp Yes Yes No Yes No Unknown
Rocq 9.0 INRIA OCaml Yes Yes Yes Yes Yes Yes
Twelf 1.7.1 Frank Pfenning, Carsten Schürmann Standard ML Yes Yes Unknown No No Unknown
  • ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition.
  • HOL theorem provers – A family of tools ultimately derived from the LCF theorem prover. In these systems, the logical core is a library of their programming language. Theorems represent new elements of the language and can only be introduced via "strategies" which guarantee logical correctness. Strategy composition gives users the ability to produce significant proofs with relatively few interactions with the system. Members of the family include:
  • IMPS, An Interactive Mathematical Proof System.[11]
  • Isabelle is an interactive theorem prover where other systems can be encoded. Isabelle/HOL is its most popular instance, whose foundation is close to that of the HOL prover. Other instances include Isabelle/ZF and Isabelle/FOL[12]. The main code-base is BSD-licensed, but the Isabelle distribution bundles many add-on tools with different licenses.
  • Jape – Java based.
  • Lean is both an interactive theorem prover and a functional, dependently-typed programming language. It is based on the calculus of inductive constructions with non-cumulative universes. Since version 4 (released in 2023), it is self-hosting. It can be used to formalise mathematics (and has a large, coherent library for formal mathematics), but also for software and hardware verification.
  • LEGO
  • Matita – A light system based on the calculus of inductive constructions.
  • MINLOG – A proof assistant based on first-order minimal logic.
  • Mizar – A proof assistant based on first-order logic, in a natural deduction style, and Tarski–Grothendieck set theory.
  • PhoX – A proof assistant based on higher-order logic which is eXtensible.
  • Prototype Verification System (PVS) – a proof language and system based on higher-order logic.
  • Rocq (formerly named Coq) – A popular interactive theorem prover based on the calculus of inductive constructions.
  • Theorem Proving System (TPS) and ETPS – Interactive theorem provers also based on simply typed lambda calculus, but based on an independent formulation of the logical theory and independent implementation.

User interfaces

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A commonly used front-end for proof assistants was the Emacs-based Proof General, developed at the University of Edinburgh. Nowadays, many provers include their own editor. Rocq includes RocqIDE, which is based on OCaml/Gtk. Isabelle includes Isabelle/jEdit, which is based on jEdit and the Isabelle/Scala infrastructure for document-oriented proof processing. More recently, Visual Studio Code extensions have been developed for Rocq,[13] Isabelle by Makarius Wenzel,[14] and for Lean 4 by the leanprover developers.[15]

Formalization extent

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Freek Wiedijk has been keeping a ranking of proof assistants by the amount of formalized theorems out of a list of 100 well-known theorems. As of September 2025, only six systems have formalized proofs of more than 70% of the theorems, namely Isabelle, HOL Light, Lean, Rocq, Metamath and Mizar.[16][17]

Notable formalized proofs

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See also: Computer-assisted proof § Theorems proved with the help of computer programs

The following is a list of notable proofs that have been formalized within proof assistants.

Theorem Proof assistant Year
Four color theorem[18] Rocq 2005
Feit–Thompson theorem[19] Rocq 2012
Fundamental group of the circle[20] Rocq 2013
Erdős–Graham problem[21][22] Lean 2022
Polynomial Freiman-Ruzsa conjecture over [23] Lean 2023
BB(5) = 47,176,870[24] Rocq 2024

See also

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References

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  1. ^ Ornes, Stephen (August 27, 2020). "Quanta Magazine – How Close Are Computers to Automating Mathematical Reasoning?".
  2. ^ Geuvers, Herman (16 July 2009). "Proof Assistants: history, ideas and future" (PDF). Sādhanā. 34: 3–25.
  3. ^ a b Paulson, Lawrence (2026-04-23). "Why not use Lean?". Retrieved 2026-04-23.
  4. ^ Boyer, Robert; Moore, J. "Proving Theorems about LISP Functions". Association for Computing Machinery. 22: 129–144.
  5. ^ Hunt, Warren; Kaufmann, Matt; Krug, Robert Bellarmine; Moore, J.; Smith, Eric W. (2005). "Meta Reasoning in ACL2" (PDF). Theorem Proving in Higher Order Logics. Lecture Notes in Computer Science. Vol. 3603. pp. 163–178. doi:10.1007/11541868_11. ISBN 978-3-540-28372-0.
  6. ^ a b c "agda/agda: Agda is a dependently typed programming language / interactive theorem prover". GitHub. Retrieved 31 July 2024.
  7. ^ "The Agda Wiki". Retrieved 31 July 2024.
  8. ^ Search for "proofs by reflection": arXiv:1803.06547
  9. ^ "Lean 4 Releases Page". GitHub. Retrieved 22 September 2025.
  10. ^ "Release v0.198 metamath/Metamath-exe". GitHub.
  11. ^ Farmer, William M.; Guttman, Joshua D.; Thayer, F. Javier (1993). "IMPS: An interactive mathematical proof system". Journal of Automated Reasoning. 11 (2): 213–248. doi:10.1007/BF00881906. S2CID 3084322. Retrieved 22 January 2020.
  12. ^ Isabelle Documentation webpage. Retrieved 22 April 2026: https://isabelle.in.tum.de/documentation.html
  13. ^ "coq-community/vscoq". July 29, 2024 – via GitHub.
  14. ^ Wenzel, Makarius. "Isabelle". Retrieved 2 November 2019.
  15. ^ "VS Code Lean 4". GitHub. Retrieved 15 October 2023.
  16. ^ Wiedijk, Freek (22 September 2025). "Formalizing 100 Theorems".
  17. ^ Geuvers, Herman (February 2009). "Proof assistants: History, ideas and future". Sādhanā. 34 (1): 3–25. doi:10.1007/s12046-009-0001-5. hdl:2066/75958. S2CID 14827467.
  18. ^ Gonthier, Georges (2008), "Formal Proof—The Four-Color Theorem" (PDF), Notices of the American Mathematical Society, 55 (11): 1382–1393, MR 2463991, archived (PDF) from the original on 2011-08-05
  19. ^ "Feit thomson proved in coq - Microsoft Research Inria Joint Centre". 2016-11-19. Archived from the original on 2016-11-19. Retrieved 2023-12-07.
  20. ^ Licata, Daniel R.; Shulman, Michael (2013). "Calculating the Fundamental Group of the Circle in Homotopy Type Theory". 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science. pp. 223–232. arXiv:1301.3443. doi:10.1109/lics.2013.28. ISBN 978-1-4799-0413-6. S2CID 5661377.
  21. ^ "Math Problem 3,500 Years In The Making Finally Gets A Solution". IFLScience. 2022-03-11. Retrieved 2024-02-09.
  22. ^ Avigad, Jeremy (2023). "Mathematics and the formal turn". arXiv:2311.00007 [math.HO].
  23. ^ Sloman, Leila (2023-12-06). "'A-Team' of Math Proves a Critical Link Between Addition and Sets". Quanta Magazine. Retrieved 2023-12-07.
  24. ^ "We have proved "BB(5) = 47,176,870"". The Busy Beaver Challenge. 2024-07-02. Retrieved 2024-07-09.

References

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Catalogues
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