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Proof assistant

December 05, 2023

For verification in computer science, see formal verification.
For the academic conference, see Interactive Theorem Proving (conference).
Not to be confused with Interactive proof system.

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.

An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right

A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics.[1]

System comparison edit

See also: Dependent type § Comparison, and Automated theorem proving § 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 No Untyped No Yes Yes[2] Already executable
Agda 2.6.3 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 8.17.0 INRIA OCaml Yes Yes Yes Yes Yes Yes
F* repository Microsoft Research and INRIA F* Yes Yes No Yes Yes[3] Yes
HOL Light repository John Harrison OCaml Yes No Yes Yes No No
HOL4 Kananaskis-13 (or repo) Michael Norrish, Konrad Slind, and others Standard ML Yes No Yes Yes No Yes
Idris 2 0.6.0. Edwin Brady Idris Yes Yes Yes Un­known Partial Yes
Isabelle Isabelle2021 (February 2021) Larry Paulson (Cambridge), Tobias Nipkow (München) and Makarius Wenzel Standard ML, Scala Yes No Yes Yes Yes Yes
Lean v4.1.0 (official release) v4.2.0-rc1 (pre-release)[4] Leonardo de Moura (Microsoft Research) C++, Lean Yes Yes Yes Yes Yes Yes
LEGO (not affiliated with Lego) 1.3.1 Randy Pollack (Edinburgh) Standard ML Yes Yes Yes No No No
Metamath v0.198[5] Norman Megill ANSI C
Mizar 8.1.05 Białystok University Free Pascal Partial Yes No No No No
Nqthm
NuPRL 5 Cornell University Common Lisp Yes Yes Yes Yes Un­known Yes
PVS 6.0 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 – 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:
  • IMPS, An Interactive Mathematical Proof System.[6]
  • 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 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. More recently, Visual Studio Code extensions have been developed for Isabelle by Makarius Wenzel,[7] and for Lean 4 by the leanprover developers.[8]

Formalization extent edit

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, Coq, Lean and Metamath.[9][10]

See also edit

Notes edit

  1. ^ Ornes, Stephen (August 27, 2020). "Quanta Magazine – How Close Are Computers to Automating Mathematical Reasoning?".
  2. ^ Hunt, Warren; Matt Kaufmann; Robert Bellarmine Krug; J Moore; Eric W. Smith (2005). "Meta Reasoning in ACL2" (PDF). Theorem Proving in Higher Order Logics. Lecture Notes in Computer Science. Vol. 3603. pp. 163–178. doi:10.1007/11541868_11. ISBN 978-3-540-28372-0.
  3. ^ Search for "proofs by reflection": arXiv:1803.06547
  4. ^ "Lean 4 Releases Page". GitHub. Retrieved 15 October 2023.
  5. ^ "Release v0.198 · metamath/Metamath-exe". GitHub.
  6. ^ 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. S2CID 3084322. Retrieved 22 January 2020.
  7. ^ Wenzel, Makarius. "Isabelle". Retrieved 2 November 2019.
  8. ^ "VS Code Lean 4". GitHub. Retrieved 15 October 2023.
  9. ^ Wiedijk, Freek (15 September 2023). "Formalizing 100 Theorems".
  10. ^ Geuvers, Herman (February 2009). "Proof assistants: History, ideas and future". Sādhanā. 34 (1): 3–25. doi:10.1007/s12046-009-0001-5. S2CID 14827467.

References edit

  • Barendregt, Henk; Geuvers, Herman (2001). (PDF). In Robinson, Alan J. A.; Voronkov, Andrei (eds.). Handbook of Automated Reasoning. Vol. 2. Elsevier. pp. 1149–. ISBN 978-0-444-50812-6. Archived from the original (PDF) on 2007-07-27.
  • Pfenning, Frank. "17. Logical frameworks" (PDF). Handbook vol 2 2001. pp. 1065–1148.
  • Pfenning, Frank (1996). "The practice of logical frameworks". In Kirchner, H. (ed.). Trees in Algebra and Programming – CAAP '96. Lecture Notes in Computer Science. Vol. 1059. Springer. pp. 119–134. doi:10.1007/3-540-61064-2_33. ISBN 3-540-61064-2.
  • Constable, Robert L. (1998). "X. Types in computer science, philosophy and logic". In Buss, S. R. (ed.). Handbook of Proof Theory. Studies in Logic. Vol. 137. Elsevier. pp. 683–786. ISBN 978-0-08-053318-6.
  • Wiedijk, Freek (2005). "The Seventeen Provers of the World" (PDF). Radboud University Nijmegen.

External links edit

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

December 05, 2023
proof, assistant, verification, computer, science, formal, verification, academic, conference, interactive, theorem, proving, conference, confused, with, interactive, proof, system, this, article, includes, list, general, references, lacks, sufficient, corresp. For verification in computer science see formal verification For the academic conference see Interactive Theorem Proving conference Not to be confused with Interactive proof system This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations November 2018 Learn how and when to remove this template message 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 An interactive proof session in CoqIDE showing the proof script on the left and the proof state on the rightA recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics 1 Contents 1 System comparison 2 User interfaces 3 Formalization extent 4 See also 5 Notes 6 References 7 External linksSystem comparison editSee also Dependent type Comparison and Automated theorem proving Comparison Name Latest version Developer s Implementation language FeaturesHigher order logic Dependent types Small kernel Proof automation Proof by reflection Code generationACL2 8 3 Matt Kaufmann and J Strother Moore Common Lisp No Untyped No Yes Yes 2 Already executableAgda 2 6 3 Ulf Norell Nils Anders Danielsson and Andreas Abel Chalmers and Gothenburg Haskell Yes Yes Yes No Partial Already executableAlbatross 0 4 Helmut Brandl OCaml Yes No Yes Yes Un known Not yet ImplementedCoq 8 17 0 INRIA OCaml Yes Yes Yes Yes Yes YesF repository Microsoft Research and INRIA F Yes Yes No Yes Yes 3 YesHOL Light repository John Harrison OCaml Yes No Yes Yes No NoHOL4 Kananaskis 13 or repo Michael Norrish Konrad Slind and others Standard ML Yes No Yes Yes No YesIdris 2 0 6 0 Edwin Brady Idris Yes Yes Yes Un known Partial YesIsabelle Isabelle2021 February 2021 Larry Paulson Cambridge Tobias Nipkow Munchen and Makarius Wenzel Standard ML Scala Yes No Yes Yes Yes YesLean v4 1 0 official release v4 2 0 rc1 pre release 4 Leonardo de Moura Microsoft Research C Lean Yes Yes Yes Yes Yes YesLEGO not affiliated with Lego 1 3 1 Randy Pollack Edinburgh Standard ML Yes Yes Yes No No NoMetamath v0 198 5 Norman Megill ANSI CMizar 8 1 05 Bialystok University Free Pascal Partial Yes No No No NoNqthmNuPRL 5 Cornell University Common Lisp Yes Yes Yes Yes Un known YesPVS 6 0 SRI International Common Lisp Yes Yes No Yes No Un knownTwelf 1 7 1 Frank Pfenning and Carsten Schurmann Standard ML Yes Yes Un known No No Un knownACL2 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 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 6 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 editA 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 Isabelle by Makarius Wenzel 7 and for Lean 4 by the leanprover developers 8 Formalization extent editFreek 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 Coq Lean and Metamath 9 10 See also editAutomated theorem proving Subfield of automated reasoning and mathematical logic Computer assisted proof Mathematical proof at least partially generated by computer Formal verification Proving or disproving the correctness of certain intended algorithms QED manifesto Proposal for a computer based database of all mathematical knowledge Satisfiability modulo theories Logical problem studied in computer science Prover9 is an automated theorem prover for first order and equational logicNotes edit Ornes Stephen August 27 2020 Quanta Magazine How Close Are Computers to Automating Mathematical Reasoning Hunt Warren Matt Kaufmann Robert Bellarmine Krug J Moore Eric W Smith 2005 Meta Reasoning in ACL2 PDF Theorem Proving in Higher Order Logics Lecture Notes in Computer Science Vol 3603 pp 163 178 doi 10 1007 11541868 11 ISBN 978 3 540 28372 0 Search for proofs by reflection arXiv 1803 06547 Lean 4 Releases Page GitHub Retrieved 15 October 2023 Release v0 198 metamath Metamath exe GitHub 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 S2CID 3084322 Retrieved 22 January 2020 Wenzel Makarius Isabelle Retrieved 2 November 2019 VS Code Lean 4 GitHub Retrieved 15 October 2023 Wiedijk Freek 15 September 2023 Formalizing 100 Theorems Geuvers Herman February 2009 Proof assistants History ideas and future Sadhana 34 1 3 25 doi 10 1007 s12046 009 0001 5 S2CID 14827467 References editBarendregt Henk Geuvers Herman 2001 18 Proof assistants using Dependent Type Systems PDF In Robinson Alan J A Voronkov Andrei eds Handbook of Automated Reasoning Vol 2 Elsevier pp 1149 ISBN 978 0 444 50812 6 Archived from the original PDF on 2007 07 27 Pfenning Frank 17 Logical frameworks PDF Handbook vol 2 2001 pp 1065 1148 Pfenning Frank 1996 The practice of logical frameworks In Kirchner H ed Trees in Algebra and Programming CAAP 96 Lecture Notes in Computer Science Vol 1059 Springer pp 119 134 doi 10 1007 3 540 61064 2 33 ISBN 3 540 61064 2 Constable Robert L 1998 X Types in computer science philosophy and logic In Buss S R ed Handbook of Proof Theory Studies in Logic Vol 137 Elsevier pp 683 786 ISBN 978 0 08 053318 6 Wiedijk Freek 2005 The Seventeen Provers of the World PDF Radboud University Nijmegen External links editThis article s use of external links may not follow Wikipedia s policies or guidelines Please improve this article by removing excessive or inappropriate external links and converting useful links where appropriate into footnote references December 2022 Learn how and when to remove this template message 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 toolsCataloguesDigital 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 Specific Logical Frameworks and Implementations DMOZ Science Math Logic and Foundations Computational Logic Logical Frameworks Retrieved from https en wikipedia org w index php title Proof assistant amp oldid 1185619274, wikipedia, wiki, book, books, library,

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