CoqCoq is an interactive theorem prover first released in 1989. It allows for expressing mathematical assertions, mechanically checks proofs of these assertions, helps find formal proofs, and extracts a certified program from the constructive proof of its formal specification. Coq works within the theory of the calculus of inductive constructions, a derivative of the calculus of constructions. Coq is not an automated theorem prover but includes automatic theorem proving tactics (procedures) and various decision procedures.
Proof assistantIn 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. A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics.
MatitaMatita is an experimental proof assistant under development at the Computer Science Department of the University of Bologna. It is a tool aiding the development of formal proofs by man-machine collaboration, providing a programming environment where formal specifications, executable algorithms and automatically verifiable correctness certificates naturally coexist. Matita is based on a dependent type system known as the Calculus of (Co)Inductive Constructions (a derivative of Calculus of Constructions), and is compatible, to some extent, with Coq.
EmacsEmacs ˈiːmæks, 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, self-documenting, real-time display editor". Development of the first Emacs began in the mid-1970s, and work on its direct descendant, GNU Emacs, is ongoing; its latest version is 29.1, released July 2023.
GNU EmacsGNU Emacs is a free software text editor. It was created by GNU Project founder Richard Stallman, based on the Emacs editor developed for Unix operating systems. GNU Emacs has been a central component of the GNU project and a flagship project of the free software movement. Its tag line is "the extensible self-documenting text editor." In 1976, Stallman wrote the first Emacs (“Editor MACroS”), and in 1984, began work on GNU Emacs, to produce a free software alternative to the proprietary Gosling Emacs.
Emacs LispEmacs Lisp is a dialect of the Lisp programming language used as a scripting language by Emacs (a text editor family most commonly associated with GNU Emacs and XEmacs). It is used for implementing most of the editing functionality built into Emacs, the remainder being written in C, as is the Lisp interpreter. Emacs Lisp is also termed Elisp, although there are also older, unrelated Lisp dialects with that name. Users of Emacs commonly write Emacs Lisp code to customize and extend Emacs.
Proof theoryProof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
MicroEMACSMicroEMACS is a small, portable Emacs-like text editor originally written by Dave Conroy in 1985, and further developed by Daniel M. Lawrence (1958–2010) and was maintained by him. MicroEMACS has been ported to many operating systems, including CP/M, MS-DOS, Microsoft Windows, VMS, Atari ST, AmigaOS, OS-9, NeXTSTEP, and various Unix-like operating systems. Variants of MicroEMACS also exist, such as mg, a more GNU Emacs-compatible editor. Many relationships to contemporary editors can also be found in MicroEMACS.
XEmacsXEmacs is a graphical- and console-based text editor which runs on almost any Unix-like operating system as well as Microsoft Windows. XEmacs is a fork, based on a version of GNU Emacs from the late 1980s. Any user can download, use, and modify XEmacs as free software available under the GNU General Public License version 2 or any later version. Between 1987 and 1993 significant delays occurred in bringing out a new version of GNU Emacs (presumed to be version 19). In the late 1980s, Richard P. Gabriel's Lucid Inc.
Agda (programming language)Agda is a dependently typed functional programming language originally developed by Ulf Norell at Chalmers University of Technology with implementation described in his PhD thesis. The original Agda system was developed at Chalmers by Catarina Coquand in 1999. The current version, originally known as Agda 2, is a full rewrite, which should be considered a new language that shares a name and tradition. Agda is also a proof assistant based on the propositions-as-types paradigm, but unlike Coq, has no separate tactics language, and proofs are written in a functional programming style.
Computer-assisted proofA computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program.
Constructive proofIn mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.
Editor warThe editor war is the rivalry between users of the Emacs and vi (now usually Vim, or more recently Neovim) text editors. The rivalry has become an enduring part of hacker culture and the free software community. The Emacs versus vi debate was one of the original "holy wars" conducted on Usenet groups, with many flame wars fought between those insisting that their editor of choice is the paragon of editing perfection, and insulting the other, since at least 1985.
Isabelle (proof assistant)The Isabelle automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring yet supporting explicit proof objects. Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories as well as implementations for code-generation, documentation, and specific support for a variety of formal methods.
Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Idris (programming language)Idris is a purely-functional programming language with dependent types, optional lazy evaluation, and features such as a totality checker. Idris may be used as a proof assistant, but is designed to be a general-purpose programming language similar to Haskell. The Idris type system is similar to Agda's, and proofs are similar to Coq's, including tactics (theorem proving functions/procedures) via elaborator reflection. Compared to Agda and Coq, Idris prioritizes management of side effects and support for embedded domain-specific languages.
Automated theorem provingAutomated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major impetus for the development of computer science. While the roots of formalised logic go back to Aristotle, the end of the 19th and early 20th centuries saw the development of modern logic and formalised mathematics.
Dependent typeIn computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifiers like "for all" and "there exists". In functional programming languages like Agda, ATS, Coq, F*, Epigram, and Idris, dependent types help reduce bugs by enabling the programmer to assign types that further restrain the set of possible implementations.
Lean (proof assistant)Lean is a theorem prover and programming language. It is based on the calculus of constructions with inductive types. The Lean project is an open-source project hosted on GitHub. It was launched by Leonardo de Moura at Microsoft Research in 2013. Lean has an interface, implemented as a Visual Studio Code extension and Language Server Protocol server, that differentiates it from other interactive theorem provers. It has native support for Unicode symbols, which can be typed using LaTeX-like sequences, such as "\times" for "×".
AutomationAutomation describes a wide range of technologies that reduce human intervention in processes, namely by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machines. Automation has been achieved by various means including mechanical, hydraulic, pneumatic, electrical, electronic devices, and computers, usually in combination. Complicated systems, such as modern factories, airplanes, and ships typically use combinations of all of these techniques.
