Proof assistant

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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.

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 5.0 Matt Kaufmann and J Strother Moore Common Lisp No Untyped No Yes Yes[1] Already executable
Agda 2.3.0 Ulf Norell (Chalmers) Haskell Yes Yes Yes No Partial Already executable
Coq 8.4 INRIA OCaml Yes Yes Yes Yes Yes Yes
HOL Light repository John Harrison OCaml Yes No Yes Yes No No
HOL4 Kananaskis-8 (or repo) Michael Norrish, Konrad Slind, and others Standard ML Yes No Yes Yes No Yes
Isabelle 2013 (or repo) Larry Paulson (Cambridge), Tobias Nipkow (München) and Makarius Wenzel (Paris-Sud) Standard ML Yes No Yes Yes Yes Yes
LEGO 1.3.1 Randy Pollack (Edinburgh) Standard ML Yes Yes Yes No No No
Mizar 8.1.02 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:
  • Isabelle is a logical framework (or generic theorem prover) that is best known for its main application: Isabelle/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[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.

See also[edit]

References[edit]

  • Frank Pfenning (1996). "The Practice of Logical Frameworks".
  • Robert L. Constable (1998). "Types in computer science, philosophy and logic". In Handbook of Proof Theory.
  • Frank Pfenning (1999). "Logical frameworks". In Handbook of Automated Reasoning.
  • H. Geuvers. "Proof assistants: History, ideas and future".
  • Freek Wiedijk. "The Seventeen Provers of the World"
  1. ^ Hunt, Warren; Matt Kaufmann, Robert Bellarmine Krug, J Moore, and Eric W. Smith (2005). "Meta Reasoning in ACL2". Springer Lecture Notes in Computer Science 3603: 163–178. 

External links[edit]

Catalogues