Proof assistant

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.

For the academic conference, see Interactive Theorem Proving (conference).
Not to be confused with Interactive proof system.
An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right.

Comparison of systems
See also: Dependent type § Comparison, and Automated theorem proving § Comparison
NameLatest versionDeveloper(s)Implementation languageFeatures
Higher-order logicDependent typesSmall kernelProof automationProof by reflectionCode generation
ACL28.2Matt Kaufmann and J Strother MooreCommon LispNoUntypedNoYesYes[1]Already executable
Agda2.6.0.1 Ulf Norell, Nils Anders Danielsson, and Andreas Abel (Chalmers and Gothenburg)HaskellYesYesYesNoPartialAlready executable
Albatross0.4 Helmut BrandlOCamlYesNoYesYesUnknownNot Yet Implemented
Coq8.11.0INRIAOCamlYesYesYesYesYesYes
F*repositoryMicrosoft Research and INRIAF*YesYesNoYesYes[2]Yes
HOL LightrepositoryJohn HarrisonOCamlYesNoYesYesNoNo
HOL4Kananaskis-12 (or repo)Michael Norrish, Konrad Slind, and othersStandard MLYesNoYesYesNoYes
IsabelleIsabelle2020 (April 2020)Larry Paulson (Cambridge), Tobias Nipkow (München) and Makarius WenzelStandard ML, ScalaYesNoYesYesYesYes
Lean v3.4.2[3] Microsoft Research C++ Yes Yes Yes Yes Yes Unknown
LEGO (not affiliated with the LEGO company)1.3.1Randy Pollack (Edinburgh)Standard MLYesYesYesNoNoNo
Mizar8.1.05Białystok UniversityFree PascalPartialYesNoNoNoNo
NuPRL5Cornell UniversityCommon LispYesYesYesYesUnknownYes
PVS6.0SRI InternationalCommon LispYesYesNoYesNoUnknown
Twelf1.7.1Frank Pfenning and Carsten SchürmannStandard MLYesYesUnknownNoNoUnknown
  • 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:
    • 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[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

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]

gollark: It's not a QR code.
gollark: No.
gollark: They *are* often kind of insecure though.
gollark: It's not just some arbitrary thing where you have to set up a GPS host or it won't tell you some location it magically got, it actually *requires* the hosts to work.
gollark: It uses true range multilateration on those to figure out the location of your client device.

See also

Notes
  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.

References
Catalogues
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