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Metamath Logo
Metamath is a formal language and an associated computer program (a proof checker) for archiving, verifying, and studying mathematical proofs. Several databases of proved theorems have been developed using Metamath covering standard results in logic, set theory, number theory, algebra, topology and analysis, among others. , the set of proved theorems using Metamath is one of the largest bodies of formalized mathematics, containing in particular proofs of 74 of the 100 theorems of the "Formalizing 100 Theorems" challenge, making it fourth after HOL Light, Isabelle, and Coq, but before Mizar, ProofPower, Lean, Nqthm, ACL2, and Nuprl. There are at least 19 proof verifiers for databases that use the Metamath format. This project is the first one of its kind that allows for interactive browsing of its formalized theorems database in the form of an ordinary website. Metamath language The Metamath language is a metalanguage, suitable for developing a wide variety of formal syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metamath Logo
Metamath is a formal language and an associated computer program (a proof checker) for archiving, verifying, and studying mathematical proofs. Several databases of proved theorems have been developed using Metamath covering standard results in logic, set theory, number theory, algebra, topology and analysis, among others. , the set of proved theorems using Metamath is one of the largest bodies of formalized mathematics, containing in particular proofs of 74 of the 100 theorems of the "Formalizing 100 Theorems" challenge, making it fourth after HOL Light, Isabelle, and Coq, but before Mizar, ProofPower, Lean, Nqthm, ACL2, and Nuprl. There are at least 19 proof verifiers for databases that use the Metamath format. This project is the first one of its kind that allows for interactive browsing of its formalized theorems database in the form of an ordinary website. Metamath language The Metamath language is a metalanguage, suitable for developing a wide variety of formal syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. It can be seen as an IDE for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle ''Archive of Formal Proofs'' (Isabelle AFP) Isabelle was named by Lawrence Paulson after Gérard Huet's daughter. The Isabelle theorem prover is free software, released under the revised BSD license. Features Isabelle is generic: it provides a meta-logic (a weak type theory), which is used to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modus Ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "''P implies Q.'' ''P'' is true. Therefore ''Q'' must also be true." ''Modus ponens'' is closely related to another valid form of argument, ''modus tollens''. Both have apparently similar but invalid forms such as affirming the consequent, denying the antecedent, and evidence of absence. Constructive dilemma is the disjunctive version of ''modus ponens''. Hypothetical syllogism is closely related to ''modus ponens'' and sometimes thought of as "double ''modus ponens''." The history of ''modus ponens'' goes back to antiquity. The first to explicitly describe the argument form ''modus ponens'' was Theophrastus. It, along with ''modus tollens'', is one of the standard patterns of inference that can be applied to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
HTML
The HyperText Markup Language or HTML is the standard markup language for documents designed to be displayed in a web browser. It can be assisted by technologies such as Cascading Style Sheets (CSS) and scripting languages such as JavaScript. Web browsers receive HTML documents from a web server or from local storage and render the documents into multimedia web pages. HTML describes the structure of a web page semantically and originally included cues for the appearance of the document. HTML elements are the building blocks of HTML pages. With HTML constructs, images and other objects such as interactive forms may be embedded into the rendered page. HTML provides a means to create structured documents by denoting structural semantics for text such as headings, paragraphs, lists, links, quotes, and other items. HTML elements are delineated by ''tags'', written using angle brackets. Tags such as and directly introduce content into the page. Other tags such as surround ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicate Calculus
{{disambiguation ...
Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, or n-ary predicate **Boolean-valued function **Syntactic predicate, in formal grammars and parsers **Functional predicate *Predication (computer architecture) *in United States law, the basis or foundation of something **Predicate crime **Predicate rules, in the U.S. Title 21 CFR Part 11 * Predicate, a term used in some European context for either nobles' honorifics or for nobiliary particles See also * Predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LF (logical Framework)
In logic, a logical framework provides a means to define (or present) a logic as a signature in a higher-order type theory in such a way that Provability logic, provability of a formula in the original logic reduces to a type inhabitation problem in the framework type theory. This approach has been used successfully for (interactive) automated theorem proving. The first logical framework was Automath; however, the name of the idea comes from the more widely known Edinburgh Logical Framework, LF. Several more recent proof tools like Isabelle (theorem prover), Isabelle are based on this idea. Unlike a direct embedding, the logical framework approach allows many logics to be embedded in the same type system. Overview A logical framework is based on a general treatment of syntax, rules and proofs by means of a dependent type theory, dependently typed lambda calculus. Syntax is treated in a style similar to, but more general than Per Martin-Löf's system of arities. To describe a logical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal System
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined abstraction, system of abstract thought. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Background Each formal system is described by primitive Symbol (formal), symbols (which collectively form an Alphabet (computer science), alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation. The system thus consists of valid formulas built up through finite combinations of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metalanguage
In logic and linguistics, a metalanguage is a language used to describe another language, often called the ''object language''. Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, quotation marks, or writing on a separate line. The structure of sentences and phrases in a metalanguage can be described by a metasyntax. Types There are a variety of recognized metalanguages, including ''embedded'', ''ordered'', and ''nested'' (or ''hierarchical'') metalanguages. Embedded An embedded metalanguage is a language formally, naturally and firmly fixed in an object language. This idea is found in Douglas Hofstadter's book, '' Gödel, Escher, Bach'', in a discussion of the relationship between formal languages and number theory: "... it is in the nature of any formalization of number theory that its metalanguage is embedded within it." It occurs in natural, or informal, languages, as well—such as in English, where words ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 to support automated reasoning in inductive logical theories, mostly for the purpose of software and hardware verification. The input language and implementation of ACL2 are written in Common Lisp. ACL2 is free and open-source software. Overview The ACL2 programming language is an applicative (side-effect free) variant of Common Lisp. ACL2 is untyped. All ACL2 functions are total — that is, every function maps each object in the ACL2 universe to another object in its universe. ACL2's base theory axiomatizes the semantics of its programming language and its built-in functions. User definitions in the programming language that satisfy a ''definitional principle'' extend the theory in a way that maintains the theory's logical cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nqthm
Nqthm is a theorem prover sometimes referred to as the Boyer–Moore theorem prover. It was a precursor to ACL2. History The system was developed by Robert S. Boyer and J Strother Moore, professors of computer science at the University of Texas, Austin. They began work on the system in 1971 in Edinburgh, Scotland. Their goal was to make a fully automatic, logic-based theorem prover. They used a variant of Pure LISP as the working logic. Definitions Definitions are formed as totally recursive functions, the system makes extensive use of rewriting and an induction heuristic that is used when rewriting and something that they called symbolic evaluation fails. The system was built on top of Lisp and had some very basic knowledge in what was called "Ground-zero", the state of the machine after bootstrapping it onto a Common Lisp implementation. This is an example of the proof of a simple arithmetic theorem. The function is part of the (called a "satellite") and is defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
