Interactive theorem proving



computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, a proof assistant or interactive theorem prover is a software tool to assist with the development of
formal proof In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequ ...
s 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
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System comparison

ACL2 ACL2 ("A Computational Logic for Applicative Common Lisp") is a software system consisting of a programming language, created by Timothy Still it was an extensible theory in a first-order logic, and an automated theorem prover. ACL2 is designed t ...
 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition. * 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
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. ** HOL Light – A thriving "minimalist fork".
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based. **ProofPower – Went proprietary, then returned to open source. Based on Standard ML. * IMPS, An Interactive Mathematical Proof System * 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 Lean, leaning or LEAN may refer to: Business practices * Lean thinking, a business methodology adopted in various fields ** Lean construction, an adaption of lean manufacturing principles to the design and construction process ** Lean governm ...
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* 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 PHOX was an American six-piece alternative folk / indie pop band from Baraboo, Wisconsin. Active from 2011 until 2017, they released their self-titled debut LP on June 24, 2014. History Prior to the formation of PHOX, many eventual members co ...
 – 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 Emacs , originally named EMACS (an acronym for "Editor MACroS"), is a family of text editors that are characterized by their extensibility. The manual for the most widely used variant, GNU Emacs, describes it as "the extensible, customizable, ...
-based Proof General, developed at the
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. Coq includes CoqIDE, which is based on OCaml/
Gtk GTK (formerly GIMP ToolKit and GTK+) is a free and open-source cross-platform widget toolkit for creating graphical user interfaces (GUIs). It is licensed under the terms of the GNU Lesser General Public License, allowing both free and proprie ...
. 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.

See also

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External links

{{external links, date=December 2022
Theorem Prover Museum

in ''Certified Programming with Dependent Types''.
Introduction to the Coq Proof Assistant
(with a general introduction to interactive theorem proving)

A list of theorem proving tools
; Catalogues

DMOZ DMOZ (from '''', an earlier domain name, stylized in lowercase in its logo) was a multilingual open-content directory of World Wide Web links. The site and community who maintained it were also known as the Open Directory ...

Science: Math: Logic and Foundations: Computational Logic: Logical Frameworks
Argument technology Automated theorem proving de:Maschinengestütztes Beweisen