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Interactive theorem prover software
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
Proof_assistant
Proof assistant and programming language
Lean is a proof assistant and a functional programming language. It is based on the calculus of constructions with inductive types. It is a free and open-source
Lean_(proof_assistant)
Higher-order logic (HOL) automated theorem prover
core (kernel) to increase the trustworthiness of proofs without requiring, yet supporting, explicit proof objects. Isabelle is available inside a flexible
Isabelle_(proof_assistant)
Interactive theorem proving systems
Nipkow; Markus Wenzel; Lawrence C. Paulson (2002). Isabelle/HOL: A Proof Assistant for Higher-Order Logic. Berlin, Heidelberg: Springer-Verlag. ISBN 978-3-540-45949-1
HOL_(proof_assistant)
Structure editor for monomorphic Martin-Löf type theory
University. It is a predecessor of the Alfa, Agda, Cayenne and Rocq proof assistants and dependently typed programming languages. It was the first language
ALF_(proof_assistant)
Proof assistant
mathematical assertions, mechanical checking of proofs of these assertions, assists in finding formal proofs using proof automation routines and extraction of a
Rocq
Functional programming language created in 2007
programming language similar to Haskell, but may also be used as a proof assistant. The Idris type system is similar to Agda's. Compared to Agda, Idris
Idris_(programming_language)
Proof assistant program
LEGO is a proof assistant developed by Randy Pollack at the University of Edinburgh. It implements several type theories: the Edinburgh Logical Framework
LEGO_(proof_assistant)
Formal argument for the existence of God
Gödel's ontological proof is a formal argument by the mathematician Kurt Gödel (1906–1978) for the existence of God. The argument is in a line of development
Gödel's_ontological_proof
Functional programming language
considered a new language that shares a name and tradition. Agda is also a proof assistant based on the propositions-as-types paradigm (Curry–Howard correspondence)
Agda_(programming_language)
Proof assistant
Matita is an experimental proof assistant under development at the Computer Science Department of the University of Bologna. It is a tool aiding the development
Matita
Reasoning for mathematical statements
that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated proof assistants, this is rarely done
Mathematical_proof
Computer scientist
computer scientist, and creator of the Z3 Theorem Prover and the Lean proof assistant during his time at Microsoft Research. He currently works at AWS and
Leonardo_de_Moura
Mathematical theory of data types
theory of types and Per Martin-Löf's intuitionistic type theory. Many proof assistants are based on type theory. For example, the underlying formal language
Type_theory
General purpose functional programming language
the pair into its three components. HOL (proof assistant) Hope (programming language) Isabelle (proof assistant) Nuprl Gordon, M.; Milner, R.; Wadsworth
ML_(programming_language)
Type theory created by Thierry Coquand
and hence consistent. The CoC has been developed alongside the Rocq proof assistant. As features were added (or possible liabilities removed) to the theory
Calculus_of_constructions
Establishment of a theorem using inference from the axioms
Mathematical proof Proof assistant Proof calculus Proof theory Proof (truth) De Bruijn factor Kassios, Yannis (February 20, 2009). "Formal Proof" (PDF). cs
Formal_proof
1995 publication in mathematics
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves
Wiles's proof of Fermat's Last Theorem
Wiles's_proof_of_Fermat's_Last_Theorem
Area of mathematics using condensed sets
and verified proof. Over a 6-month period, a group led by Johan Commelin verified the central part of the proof using the proof assistant Lean. As of 14
Condensed_mathematics
Subfield of computer science and logic
essentially another proof assistant for classical higher order logic. Rocq Developed in France, Rocq is another automated proof assistant, which can automatically
Automated_reasoning
1970s automated theorem prover
HOL, HOL Light, and the Isabelle proof assistant that supports various logics. As of 2019, the Isabelle proof assistant still contains an implementation
Logic for Computable Functions
Logic_for_Computable_Functions
Topics referred to by the same term
Lean startup, how to start a company in a lean way Lean (proof assistant), a proof assistant and programming language Lean (drug), based on cough syrup
Lean
Proof assistant program
for writing mathematical definitions and proofs, a proof assistant, which is able to mechanically check proofs written in this language, and a library
Mizar_system
Type theory in logic and mathematics
make possible); and the formalization of each of these in computer proof assistants. There is a large overlap between the work referred to as homotopy
Homotopy_type_theory
Formal language and associated computer program
language and an associated computer program (a proof assistant) for archiving and verifying mathematical proofs. Several databases of proved theorems have
Metamath
proved in January 2026 by Enrique Barschkis; the proof was formally verified in the Lean proof assistant. In March 2026, Erdős problem 369 was solved by
List of conjectures by Paul Erdős
List_of_conjectures_by_Paul_Erdős
Proof assistant program
Jape is a configurable, graphical proof assistant, originally developed by Richard Bornat at Queen Mary, University of London and Bernard Sufrin the University
Jape_(software)
Software used in mathematical applications
Mathematical software is software used to model, analyze or calculate numeric, symbolic or geometric data. Numerical analysis and symbolic computation
Mathematical_software
Formally verified C compiler
Automation (INRIA). The compiler is specified, programmed and proven in proof assistant software named Rocq. CompCert is to be used to program embedded systems
CompCert
Software for automated theorem proving
IsaPlanner is a proof planner for the interactive proof assistant, Isabelle, originally developed by Lucas Dixon. IsaPlanner 2: A Proof Planner in Isabelle
IsaPlanner
Line-by-line system for natural deduction proofs
2025. "The Jape general-purpose proof assistant". GitHub. Retrieved 6 May 2025. (see Jape) "Resources for typesetting proofs in Fitch notation with LaTeX"
Fitch_notation
Programming language
Dafny also allows the proof of properties between one function and another. As is common for a proof assistant, such proofs are often inductive in nature
Dafny
Analysis of computer programs without executing them
Union-find Tools Constraint solvers CHC SAT SMT Lightweight Alloy TLA+ Proof assistants ACL2 Agda F* HOL Light HOL4 Idris Isabelle Isabelle/HOL Lean LEGO Mizar
Static_program_analysis
Mathematical proof at least partially generated by computer
new proofs for known theorems.[citation needed] Additionally, interactive proof assistants allow mathematicians to develop human-readable proofs which
Computer-assisted_proof
Class of mathematical expression
by zero, which can be handled in another part of the program. Many proof assistants, such as Rocq and Lean, define 1/0 = 0. This is to make all functions
Division_by_zero
electronic programs that perform proof assistance with theorems in mathematical logic. In the 21st century, these proof assistant programs have given birth to
Logical_machine
Data structure for storing non-overlapping sets
disjoint-set forest data structure and formalized its correctness using the proof assistant Rocq (then: Coq). "Semi-persistent" means that previous versions of
Disjoint-set_data_structure
Subfield of automated reasoning and mathematical logic
consequently more easily understandable and checkable, have been developed. Proof assistants require a human user to give hints to the system. Depending on the
Automated_theorem_proving
Functional programming language inspired by ML and aimed at program verification
combination of satisfiability modulo theories (SMT) solving and manual proofs. For execution, programs written in F* can be translated to OCaml, F#, C
F*_(programming_language)
Concept in computer science
interactive theorem provers such as Rocq (previously known as Coq) and HOL (proof assistant). In comparison to the program analysis work, these tools require more
Separation_logic
French computer scientist and mathematician
(formerly named Coq, the name is a reference partly to Coquand's surname) proof assistant, which he began developing in 1984 while working at INRIA (a French
Thierry_Coquand
Sub-field of artificial intelligence
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
Argument_technology
Topics referred to by the same term
one of the álfar alf, ISO-639-3 code for the Alege language ALF (proof assistant), dependently typed programming language/theorem prover Action Language
ALF
Number in {..., –2, –1, 0, 1, 2, ...}
or pair(2,2), etc.. This technique of construction is used by the proof assistant Isabelle; however, many other tools use alternative construction techniques
Integer
Canadian computer scientist
four color theorem and Feit–Thompson proof of the odd-order theorem. (Both were written using the proof assistant Rocq.) In 2011, as a principal researcher
Georges_Gonthier
Relationship between programs and proofs
language theory and proof theory, the Curry–Howard correspondence is a direct relationship between computer programs and mathematical proofs. It is also known
Curry–Howard_correspondence
Proof assistant program
MINLOG is a proof assistant developed at LMU Munich by the team of Helmut Schwichtenberg. MINLOG is based on first order natural deduction calculus. It
MINLOG
Topics referred to by the same term
villages), mines, and dams Locale (Isabelle), a module of the Isabelle proof assistant Locale ('Ndrangheta), the main local organizational unit of the 'Ndrangheta
Locale
Proof assistant program
In automated theorem proving, PhoX is a proof assistant based on higher-order logic which is eXtensible. The user gives PhoX an initial goal and guides
PhoX
incorrect (or no) proofs? Moritz. Theorems demoted back to conjectures Mei Zhang, Proofs shown to be wrong after formalization with proof assistant Steven-Owen
List_of_incomplete_proofs
Functional programming language
dependent types bring may be found in The Epigram Tutorial. ALF, a proof assistant among the predecessors of Epigram. McBride, Conor; McKinna, James (2004)
Epigram (programming language)
Epigram_(programming_language)
Of a function, an additional effect besides returning a value
Union-find Tools Constraint solvers CHC SAT SMT Lightweight Alloy TLA+ Proof assistants ACL2 Agda F* HOL Light HOL4 Idris Isabelle Isabelle/HOL Lean LEGO Mizar
Side effect (computer science)
Side_effect_(computer_science)
Programming paradigm restricted to provably terminating programs
Total functional programming (also known as strong functional programming, to be contrasted with ordinary, or weak functional programming) is a programming
Total_functional_programming
Formula for area of a grid polygon
of different proof assistants. As of 2024[update], Pick's theorem had been formalized and proven in only two of the ten proof assistants recorded by Wiedijk
Pick's_theorem
Computer program for the Boolean satisfiability problem
used to assist in proving mathematical theorems through computer-assisted proof. In Ramsey theory, several previously unknown Van der Waerden numbers were
SAT_solver
Italian computer scientist
a non-profit organization supporting the development of the Lean proof assistant and its ecosystem. Mathematics Genealogy Project https://www.phasecraft
Simone_Severini
Rules to verify computer program correctness
deficiency in the axioms and rules quoted above is that they give no basis for a proof that a program successfully terminates. Failure to terminate may be due
Hoare_logic
Dynamic window manager for X Window System
projects. This has included re-implementing xmonad using the Rocq proof assistant, a determination xmonad is an imperative program, and studies of package
Xmonad
Proof assistant program
HOL Light is a proof assistant for classical higher-order logic. It is a member of the HOL theorem prover family. Compared with other HOL systems, HOL
HOL_Light
improve the proofs generated by automated theorem proving tools such as SAT solvers, SMT-solvers, first-order theorem provers and proof assistants. In propositional
Proof_compression
Free and open-source VPN protocol
INRIA published a machine-checked proof of the WireGuard protocol, produced using the CryptoVerif proof assistant. WireGuard supports pre-shared symmetric
WireGuard
Data type in type theory
quotient types were defined and implemented as part of the Nuprl proof assistant, in work led by Robert L. Constable and others. Quotient types have
Quotient_type
Programming language and theorem prover
Centaur Technology, IBM, Intel, Oracle, and Collins Aerospace. List of proof assistants "XDOC — Note-1-7". www.cs.utexas.edu. "ACM: Press Release, March 15
ACL2
Topics referred to by the same term
during the Iraq war Sledgehammer (Isabelle), a tool of the Isabelle proof assistant Sledgehammer (ride), a ride at Canada's Wonderland theme park Sledgehammer
Sledgehammer_(disambiguation)
Proof assistant program
under the GNU General Public License (GPL). Formal methods List of proof assistants Rosetta-lang Owre, Shankar, and Rushby, 1992. PVS: A Prototype Verification
Prototype_Verification_System
theories have semantics in various types of cubical sets. The Agda proof assistant includes an implementation of cubical type theory. Cubical type theory
Cubical_type_theory
German computer scientist (born 1958)
interactive and automatic theorem proving, in particular for the Isabelle proof assistant; he was the editor of the Journal of Automated Reasoning up to January
Tobias_Nipkow
Software projects developed at universities
Mizar – proof assistant and formal mathematics system (Białystok, Alberta, and Shinshu) Murφ – explicit-state model checker (Stanford) Nuprl – proof development
List of software developed at universities
List_of_software_developed_at_universities
Field of knowledge
computers and proof assistants. Even with this modern technology, it may take years of human work for writing down a completely detailed proof. This does
Mathematics
Decidable first-order theory of the natural numbers with addition
example, the Rocq and Lean proof assistant systems feature the tactic omega for Presburger arithmetic and the Isabelle proof assistant contains a verified quantifier
Presburger_arithmetic
Classification theorem in group theory
more leisurely style. A fully formal proof, checked with the Rocq proof assistant, was announced in September 2012 by Georges Gonthier and fellow researchers
Feit–Thompson_theorem
Controlled language
been used in the fields of software specifications, theorem proving, proof assistants, text summaries, ontologies, rules, querying, medical documentation
Attempto_Controlled_English
Limitative results in mathematical logic
completely verified by proof assistant software. Gödel's original proofs of the incompleteness theorems, like most mathematical proofs, were written in natural
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Topics referred to by the same term
Hol, a community in Duk County in Jonglei State in South Sudan HOL (proof assistant), a family of interactive theorem proving systems Head-of-line blocking
Hol
Computer science field
the violation is feasible, it is reported to the user. If it is not, the proof of infeasibility is used to refine the abstraction and checking begins again
Model_checking
constructions in 1984 with Thierry Coquand. He led the Coq (renamed Rocq) proof assistant project in the 1990s with Christine Paulin-Mohring, who developed Coq
Gérard_Huet
Math theorem about sphere packing
of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants. In 2017, the formal proof was accepted
Kepler_conjecture
Theorem classifying finite simple groups
group of Lie type, in which case there would be 27 sporadic groups). The proof consists of tens of thousands of pages in several hundred journal articles
Classification of finite simple groups
Classification_of_finite_simple_groups
Proposal for a computer-based database of all mathematical knowledge
proof of the irrationality of the square root of two in seventeen different proof assistants. Formalized Mathematics a journal in which Mizar proofs are
QED_manifesto
Planar maps require at most four colors
This proof remains unpublished, however. In 2005, Benjamin Werner and Georges Gonthier formalized a proof of the theorem inside the Coq proof assistant. This
Four_color_theorem
Computer programming concept
complex constraints can be formally verified interactively with a proof assistant. Preconditions in object-oriented software development are an essential
Precondition
Programming language
types. The Haxe programming language compiler. HOL Light, a formal proof assistant. Infer, a static analyzer created at Facebook for Java, C, C++, and
OCaml
Fundamental problem in computer science
Union-find Tools Constraint solvers CHC SAT SMT Lightweight Alloy TLA+ Proof assistants ACL2 Agda F* HOL Light HOL4 Idris Isabelle Isabelle/HOL Lean LEGO Mizar
Path_explosion
Basic framework of mathematics
theory have also gained prevalence, being commonly used in computer proof assistants. It results from this that the basic mathematical concepts, such as
Foundations_of_mathematics
Area of mathematics
theorems (for example the four color theorem), and the design and use of proof assistants. Computational mathematics emerged as a distinct part of applied mathematics
Computational_mathematics
Functional programming language
clearer error messages by disabling type classes as a default. Agda is a proof assistant written in Haskell. Cabal is a tool for building and packaging Haskell
Haskell
Topics referred to by the same term
processing JAPE, an automated pun generator Jape (software), a Java-based proof assistant Java Annotations Pattern Engine – see General Architecture for Text
Jape
Union-find Tools Constraint solvers CHC SAT SMT Lightweight Alloy TLA+ Proof assistants ACL2 Agda F* HOL Light HOL4 Idris Isabelle Isabelle/HOL Lean LEGO Mizar
Hyperproperty
Formal languages for expressing mathematical theories
very influential in the later development of logical frameworks and proof assistants. The Mizar system, a system of writing and checking formalized mathematics
Automath
Type of Turing machine
them to formally prove the correctness of the machine in the Matita proof assistant. Alternating Turing machine – Abstract computation model Counter machine –
Universal_Turing_machine
Method of forming inferences
goal. It is used in automated theorem provers, inference engines, proof assistants, and other artificial intelligence applications. In game theory, researchers
Backward_chaining
Methods in artificial intelligence research
to be useful in a few domains such as computer algebra systems and proof assistants. A short history of symbolic AI to the present day follows below. Time
Symbolic artificial intelligence
Symbolic_artificial_intelligence
Branch of computer science
substantial subset of Standard ML programming language using the HOL proof assistant. Another approach to obtain a formally correct compiler is to use semantics-directed
Compiler_correctness
Association for Computing Machinery (ACM) special interest group
Prover 2014: GNU Compiler Collection (GCC) 2013: Rocq (then: Coq) proof assistant 2012: Jikes Research Virtual Machine (RVM) awarded to Bowen Alpern
SIGPLAN
Process of analyzing computer program behavior
Union-find Tools Constraint solvers CHC SAT SMT Lightweight Alloy TLA+ Proof assistants ACL2 Agda F* HOL Light HOL4 Idris Isabelle Isabelle/HOL Lean LEGO Mizar
Program_analysis
Union-find Tools Constraint solvers CHC SAT SMT Lightweight Alloy TLA+ Proof assistants ACL2 Agda F* HOL Light HOL4 Idris Isabelle Isabelle/HOL Lean LEGO Mizar
Dependence_analysis
Mathematics independent of applications
inductive process). Then, putative proofs generated by artificial intelligence can be verified via proof assistant tools such as Lean, and large supporting
Pure_mathematics
Proving or disproving the correctness of certain intended algorithms
verification of these systems is done by ensuring the existence of a formal proof of a mathematical model of the system. Examples of mathematical objects
Formal_verification
Algorithm for computing Gröbner bases
implementation of Buchberger’s algorithm has been proved correct within the proof assistant Coq (renamed Rocq). In the SymPy library for Python, the (improved)
Buchberger's_algorithm
Formal system of logic
Type theory Higher-order grammar Higher-order logic programming HOL (proof assistant) Many-sorted logic Typed lambda calculus Modal logic Jacobs, 1999,
Higher-order_logic
PROOF ASSISTANT
PROOF ASSISTANT
Boy/Male
Indian
Proof
Girl/Female
Indian
Witness; Proof
Boy/Male
Arabic
Proof; Evidence
Boy/Male
Muslim
Proof
Boy/Male
Muslim
Evidence. Proof.
Girl/Female
Muslim
Proof
Boy/Male
Arabic, Muslim
The Proof
Girl/Female
Muslim/Islamic
Guide Proof
Boy/Male
Muslim
Proof
Boy/Male
Muslim/Islamic
Proof
Boy/Male
Arabic, French, German, Gujarati, Hindu, Indian, Malaysian, Muslim, Turkish
Proof; Evidence
Boy/Male
Indian
Proof
Surname or Lastname
English
English : variant of Rolfe.German : from Ruffo, a short form of a personal name formed with hrÅd ‘renown’, ‘victory’.Probably an Americanized spelling of German Ruf and Ruff.
Boy/Male
Arabic, Muslim
Evidence; Proof
Girl/Female
Muslim
Guide, Proof
Boy/Male
Muslim/Islamic
Proof
Boy/Male
Muslim
Argument, Reasoning, Proof
Boy/Male
Afghan, Arabic, Hindu, Indian, Muslim
Proof
Girl/Female
Arabic, Muslim
Guide; Proof
Boy/Male
Indian
Argument, Reasoning, Proof
PROOF ASSISTANT
PROOF ASSISTANT
Girl/Female
Arabic, Muslim
Adornment of Women
Boy/Male
Tamil
Lord Krishna
Girl/Female
Irish French
Beautiful.
Girl/Female
Arabic, Egyptian, French, Muslim
To Collect; To Rest; Married to the Prophet; Wife of Prophet Muhammad; Gathering; Brave
Girl/Female
Finnish, German
One who Pour Water from a Jug
Boy/Male
Sikh
Longing for beloved God
Female
Chinese
morning sunlight.
Biblical
master; lord,Lord,"owner" or "lord", also "husband" (as possessor of the wife);possessor, controller;
Surname or Lastname
English (Lancashire)
English (Lancashire) : habitational name from a hamlet near Parbold, not far from Wigan, so named from Old English fæger ‘beautiful’ + hyrst ‘wooded hill’.
Female
Scottish
 Pet form of Scottish Jean, JESSIE means "God is gracious." Compare with other forms of Jessie.
PROOF ASSISTANT
PROOF ASSISTANT
PROOF ASSISTANT
PROOF ASSISTANT
PROOF ASSISTANT
a.
Firm or successful in resisting; as, proof against harm; waterproof; bombproof.
v. t.
Armor of excellent or tried quality, and deemed impenetrable; properly, armor of proof.
a.
Proof against proofs; obstinate in the wrong.
n.
Demonstration; proof.
v. t.
To cover with a roof.
n.
Proof.
n.
Proof.
n.
Proof.
n.
A trial impression, as from type, taken for correction or examination; -- called also proof sheet.
n.
That which resembles, or corresponds to, the covering or the ceiling of a house; as, the roof of a cavern; the roof of the mouth.
a.
Used in proving or testing; as, a proof load, or proof charge.
n.
The cover of any building, including the roofing (see Roofing) and all the materials and construction necessary to carry and maintain the same upon the walls or other uprights. In the case of a building with vaulted ceilings protected by an outer roof, some writers call the vault the roof, and the outer protection the roof mask. It is better, however, to consider the vault as the ceiling only, in cases where it has farther covering.
n.
Trial; proof.
a.
Highly rectified; very strongly alcoholic; as, high-proof spirits.
v. t.
To arm with proof armor; to arm securely; as, to proof-arm herself.
n.
Proof.
n.
Proof; evidence.
n.
Proof; trial.