Proof assistant

File:CoqProofOfDecidablityOfEqualityOnNaturalNumbers.png

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.

History

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.

System 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	Already executable
Agda	2.6.4.3	Ulf Norell, Nils Anders Danielsson, and Andreas Abel (Chalmers and Gothenburg)	Haskell	Yes	Yes	Yes	No	Partial	Already executable
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	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	v4.23.0	Leonardo de Moura (Microsoft Research)	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	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.
• Rocq (former name: Coq) – Allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification.
• 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:
  • HOL4 – The "primary descendant", still under active development. Support for both Moscow ML and Poly/ML. Has a BSD-style license.
  • HOL Light – A thriving "minimalist fork". OCaml based.
  • ProofPower – Went proprietary, then returned to open source. Based on Standard ML.
• IMPS, An Interactive Mathematical Proof System.
• Isabelle is an interactive theorem prover, successor of HOL. The main code-base is BSD-licensed, but the Isabelle distribution bundles many add-on tools with different licenses.
• Jape – Java based.
• Lean
• 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.
• 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

A popular front-end for proof assistants is the Emacs-based Proof General, developed at the University of Edinburgh.

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, Isabelle by Makarius Wenzel, and for Lean 4 by the leanprover developers.

Formalization extent

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.

Notable formalized proofs

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

Theorem	Proof assistant	Year
Four color theorem	Rocq	2005
Feit–Thompson theorem	Rocq	2012
Fundamental group of the circle	Rocq	2013
Erdős–Graham problem	Lean	2022
Polynomial Freiman-Ruzsa conjecture over ${\mathbb{𝔽}}_2$	Lean	2023
BB(5) = 47,176,870	Rocq	2024

See also

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• Prover9 – is an automated theorem prover for first-order and equational logic

External links

• Theorem Prover Museum
• "Introduction" in Certified Programming with Dependent Types.
• Introduction to the Coq Proof Assistant (with a general introduction to interactive theorem proving)
• Interactive Theorem Proving for Agda Users
• A list of theorem proving tools

Catalogues

• Digital Math by Category: Tactic Provers
• Automated Deduction Systems and Groups
• Theorem Proving and Automated Reasoning Systems
• Database of Existing Mechanized Reasoning Systems
• NuPRL: Other Systems
• (By Frank Pfenning).
• DMOZ: Science: Math: Logic and Foundations: Computational Logic: Logical Frameworks