Proof assistant

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An interactive proof session in CoqIDE, 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.

Comparison of systems[edit]

Name Latest version Developer(s) Implementation language Features
Higher-order logic Dependent types Small kernel Proof automation Proof by reflection Code generation
ACL2 8.2 Matt Kaufmann and J Strother Moore Common Lisp No Untyped No Yes Yes[1] Already executable
Agda 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
Coq 8.11.0 INRIA OCaml Yes Yes Yes Yes Yes Yes
F* repository Microsoft Research and INRIA F* Yes Yes No Yes Yes[2] 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
Isabelle Isabelle2020 (April 2020) Larry Paulson (Cambridge), Tobias Nipkow (München) and Makarius Wenzel Standard ML, Scala Yes No Yes Yes Yes Yes
Lean v3.4.2[3] Microsoft Research C++ Yes Yes Yes Yes Yes Unknown
LEGO (not affiliated with the LEGO company) 1.3.1 Randy Pollack (Edinburgh) Standard ML Yes Yes Yes No No No
Mizar 8.1.05 Białystok University Free Pascal Partial Yes No No No No
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
Twelf 1.7.1 Frank Pfenning and 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.
  • Coq – Which 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:
  • IMPS, An Interactive Mathematical Proof System[4]
  • 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.
  • 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.
  • 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.
  • Typelab
  • Yarrow

The Theorem Prover Museum is an initiative to conserve the sources of theorem prover systems for future analysis, since they are important cultural/scientific artefacts. It has the sources of many of the systems mentioned above.

User interfaces[edit]

A popular front-end for proof assistants is the Emacs-based Proof General, developed at the University of Edinburgh. Coq includes CoqIDE, 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, a Visual Studio Code extension for Isabelle has also been developed by Makarius Wenzel.[5]

See also[edit]


  1. ^ Hunt, Warren; Matt Kaufmann; Robert Bellarmine Krug; J Moore; Eric W. Smith (2005). "Meta Reasoning in ACL2" (PDF). Springer Lecture Notes in Computer Science. 3603: 163–178.
  2. ^ Search for "proofs by reflection": arXiv:1803.06547
  3. ^ "Lean Theorem Prover Releases page". GitHub.
  4. ^ 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. Retrieved 22 January 2020.
  5. ^ Wenzel, Makarius. "Isabelle". Retrieved 2 November 2019.


External links[edit]