Proof assistant

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

Name	Latest version	Developer(s)	Implementation language	Features
Name	Latest version	Developer(s)	Implementation language	Higher-order logic	Dependent types	Small kernel	Proof automation	Proof by reflection	Code generation
ACL2	7.1	Matt Kaufmann and J Strother Moore	Common Lisp	No	Untyped	No	Yes	Yes^[1]	Already executable
Agda	2.5.1.1	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	8.8	INRIA	OCaml	Yes	Yes	Yes	Yes	Yes	Yes
F*	repository	Microsoft Research and INRIA	F*	Yes	Yes	No	Yes	Unknown	Yes
HOL Light	repository	John Harrison	OCaml	Yes	No	Yes	Yes	No	No
HOL4	Kananaskis-12 (or repo)	Michael Norrish, Konrad Slind, and others	Standard ML	Yes	No	Yes	Yes	No	Yes
Isabelle	2018	Larry Paulson (Cambridge), Tobias Nipkow (München) and Makarius Wenzel (Paris-Sud)	Standard ML, Scala	Yes	No	Yes	Yes	Yes	Yes
Lean	repository	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	5.0	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:
- 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.
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

User interface

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.

Notes

↑ 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.

References

Henk Barendregt and Herman Geuvers (2001). "Proof-assistants using Dependent Type Systems". In Handbook of Automated Reasoning.
Frank Pfenning (2001). "Logical frameworks". In Handbook of Automated Reasoning.
Frank Pfenning (1996). "The Practice of Logical Frameworks".
Robert L. Constable (1998). "Types in computer science, philosophy and logic". In Handbook of Proof Theory.
H. Geuvers. "Proof assistants: History, ideas and future".
Freek Wiedijk. "The Seventeen Provers of the World"

External links

"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

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[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.