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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
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)
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)
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)
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
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
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)
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
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
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
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
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
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
Topics referred to by the same term
startup, how to start a company in a lean way Lean (proof assistant), a mathematical proof assistant and programming language Lean (drug), based on cough
Lean
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
Sequence of operations for a task
"The Spirit of Truth" for a history leading to, and a discussion of, his proof. Kleene, Stephen C. (1936). "General Recursive Functions of Natural Numbers"
Algorithm
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
Controlled language
been used in the fields of software specifications, theorem proving, proof assistants, text summaries, ontologies, rules, querying, medical documentation
Attempto_Controlled_English
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
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
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)
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
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
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)
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
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
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
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
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
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
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
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
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
Study of discrete mathematical structures
it can be easily verified with a truth table. The study of mathematical proof is particularly important in logic, and has accumulated to automated theorem
Discrete_mathematics
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
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
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
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
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
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
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
Software used in mathematical applications
Statistical software Numerical-analysis software Numerical libraries Proof assistants Solvers Discrete Computer algebra Computational number theory Combinatorics
Mathematical_software
Theory of getting acceptably close inexact mathematical calculations
Statistical software Numerical-analysis software Numerical libraries Proof assistants Solvers Discrete Computer algebra Computational number theory Combinatorics
Approximation_theory
Computer programming concept
complex constraints can be formally verified interactively with a proof assistant. Preconditions in object-oriented software development are an essential
Precondition
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
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
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
Branch of mathematics concerning probability
both discrete and continuous distributions as well as others; separate proofs are not required for discrete and continuous distributions. Certain random
Probability_theory
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
French computer scientist
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
Application of mathematical methods to other fields
Statistical software Numerical-analysis software Numerical libraries Proof assistants Solvers Discrete Computer algebra Computational number theory Combinatorics
Applied_mathematics
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
Graph data structure
known as non-chronological backtracking), e-graphs are extended to produce proof certificates. E-graphs are also used in the Simplify theorem prover of ESC/Java
E-graph
Incubator for philanthropic ventures
venture capital and transform into startups. Development for Lean (proof assistant) is supported by a FRO. E11 bio has a goal of creating a single-cell
Convergent_Research
Study of abstract machines and automata
language. The pumping lemma for regular languages, also useful in regularity proofs, was proven in this period by Michael O. Rabin and Dana Scott, along with
Automata_theory
Branch of applied probability theory
Statistical software Numerical-analysis software Numerical libraries Proof assistants Solvers Discrete Computer algebra Computational number theory Combinatorics
Decision_theory
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
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
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
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
Theory of subatomic structure
a1. S2CID 119684549. Duncan, John; Griffin, Michael; Ono, Ken (2015). "Proof of the Umbral Moonshine Conjecture". Research in the Mathematical Sciences
String_theory
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
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
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
PROOF ASSISTANT
PROOF ASSISTANT
Girl/Female
Indian
Witness; Proof
Boy/Male
Indian
Proof
Girl/Female
Arabic, Muslim
Guide; Proof
Boy/Male
Muslim
Argument, Reasoning, Proof
Boy/Male
Indian
Argument, Reasoning, 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.
Girl/Female
Muslim/Islamic
Guide Proof
Boy/Male
Arabic
Proof; Evidence
Boy/Male
Muslim/Islamic
Proof
Girl/Female
Muslim
Proof
Boy/Male
Muslim
Proof
Boy/Male
Arabic, Muslim
The Proof
Boy/Male
Arabic, Muslim
Evidence; Proof
Girl/Female
Muslim
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
Muslim
Evidence. Proof.
Boy/Male
Indian
Proof
Boy/Male
Afghan, Arabic, Hindu, Indian, Muslim
Proof
PROOF ASSISTANT
PROOF ASSISTANT
Boy/Male
Tamil
The Moon
Boy/Male
Hindu, Indian
Son of Lord Varun
Girl/Female
Tamil
A Goddess
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Telugu
Immortal
Boy/Male
Hindu, Indian, Marathi, Tamil
Another Name for Lord Vishnu
Boy/Male
Tamil
Digvijay | திகà¯à®µà®¿à®œà®¯
Who is victorious over everyone
Boy/Male
Scottish
Speckled.
Boy/Male
Hindu, Indian, Sanskrit
Capable of Achieving Everything
Boy/Male
Tamil
Lord Indra
Girl/Female
Arabic
Jewel; Ornament
PROOF ASSISTANT
PROOF ASSISTANT
PROOF ASSISTANT
PROOF ASSISTANT
PROOF ASSISTANT
v. t.
To arm with proof armor; to arm securely; as, to proof-arm herself.
v. t.
To cover with a roof.
n.
Proof.
n.
Proof.
n.
Trial; proof.
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.
n.
Proof.
n.
Proof; trial.
a.
Proof against proofs; obstinate in the wrong.
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.
v. t.
Armor of excellent or tried quality, and deemed impenetrable; properly, armor of proof.
n.
Proof.
a.
Used in proving or testing; as, a proof load, or proof charge.
n.
Proof; evidence.
a.
Highly rectified; very strongly alcoholic; as, high-proof spirits.
n.
Demonstration; proof.
a.
Firm or successful in resisting; as, proof against harm; waterproof; bombproof.
n.
A trial impression, as from type, taken for correction or examination; -- called also proof sheet.