Proof assistant

File:CoqProofOfDecidablityOfEqualityOnNaturalNumbers.png

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.

A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Template:AnchorSystem 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 and J Strother Moore	Common Lisp	Template:No	Template:N/a	Template:No	Template:Yes	Template:Yes<ref>Template:Cite book</ref>	Template:N/a
Agda	2.6.4.3<ref name="github-agda" />	Ulf Norell, Nils Anders Danielsson, and Andreas Abel (Chalmers and Gothenburg)<ref name="github-agda" />	Haskell<ref name="github-agda" />	Template:Yes Template:Cn	Template:Yes <ref name="agdaWiki" />	Template:Yes Template:Cn	Template:No Template:Cn	Template:Partial Template:Cn	Template:N/a Template:Cn
Albatross	0.4	Helmut Brandl	OCaml	Template:Yes	Template:No	Template:Yes	Template:Yes	Template:Unknown	Template:Not yet Implemented
Rocq (formerly known as Coq)	9.0	INRIA	OCaml	Template:Yes	Template:Yes	Template:Yes	Template:Yes	Template:Yes	Template:Yes
F*	repository	Microsoft Research and INRIA	F*	Template:Yes	Template:Yes	Template:No	Template:Yes	Template:Yes<ref>Search for "proofs by reflection": Template:ArXiv</ref>	Template:Yes
HOL Light	repository	John Harrison	OCaml	Template:Yes	Template:No	Template:Yes	Template:Yes	Template:No	Template:No
HOL4	Kananaskis-13 (or repo)	Michael Norrish, Konrad Slind, and others	Standard ML	Template:Yes	Template:No	Template:Yes	Template:Yes	Template:No	Template:Yes
Idris	2 0.6.0.	Edwin Brady	Idris	Template:Yes	Template:Yes	Template:Yes	Template:Unknown	Template:Partial	Template:Yes
Isabelle	Isabelle2024 (May 2024)	Larry Paulson (Cambridge), Tobias Nipkow (München) and Makarius Wenzel	Standard ML, Scala	Template:Yes	Template:No	Template:Yes	Template:Yes	Template:Yes	Template:Yes
Lean	citation	CitationClass=web }}</ref>	Leonardo de Moura (Microsoft Research)	C++, Lean	Template:Yes	Template:Yes	Template:Yes	Template:Yes	Template:Yes	Template:Yes
LEGO	1.3.1	Randy Pollack (Edinburgh)	Standard ML	Template:Yes	Template:Yes	Template:Yes	Template:No	Template:No	Template:No
Metamath	citation	CitationClass=web }}</ref> || Norman Megill || ANSI C || || || || || ||
Mizar	8.1.11	Białystok University	Free Pascal	Template:Partial	Template:Yes	Template:No	Template:No	Template:No	Template:No
Nqthm
NuPRL	5	Cornell University	Common Lisp	Template:Yes	Template:Yes	Template:Yes	Template:Yes	Template:Unknown	Template:Yes
PVS	6.0	SRI International	Common Lisp	Template:Yes	Template:Yes	Template:No	Template:Yes	Template:No	Template:Unknown
Twelf	1.7.1	Frank Pfenning and Carsten Schürmann	Standard ML	Template:Yes	Template:Yes	Template:Unknown	Template:No	Template:No	Template: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 (formerly known as 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.<ref>Template:Cite journal</ref>
• 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.
• 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.

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, Visual Studio Code extensions have been developed for Coq,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Isabelle by Makarius Wenzel,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and for Lean 4 by the leanprover developers.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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 2023, only five systems have formalized proofs of more than 70% of the theorems, namely Isabelle, HOL Light, Rocq, Lean, and Metamath.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite journal</ref>

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<ref>Template:Citation</ref>	Coq	2005
Feit–Thompson theorem<ref>{{#invoke:citation/CS1|citation	CitationClass=web }}</ref> || Coq || 2012
Fundamental group of the circle<ref>Template:Cite book</ref>	Coq	2013
Erdős–Graham problem<ref>{{#invoke:citation/CS1|citation	CitationClass=web }}</ref><ref>Template:Cite arXiv</ref>	Lean	2022
citation	CitationClass=web }}</ref> || Lean || 2023
BB(5) = 47,176,870<ref>{{#invoke:citation/CS1|citation	CitationClass=web }}</ref>	Coq	2024

See also

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

References

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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
• {{#invoke:citation/CS1|citation

|CitationClass=web }} (By Frank Pfenning).

• DMOZ: Science: Math: Logic and Foundations: Computational Logic: Logical Frameworks