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Interactive theorem proving

 

 
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Interactive theorem proving
 
 
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Interactive theorem proving is the field of computer science
Computer science

Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems....
 and mathematical logic
Mathematical logic

Mathematical logic is a subfield of mathematics and logic with close connections to computer science and philosophical logic. The field includes the mathematical study of logic and the applications of formal logic to other areas of mathematics....
 concerned with tools to develop formal proof
Formal proof

A formal proof or derivation is a finite sequence of proposition each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference....
s by man-machine collaboration. This involves some sort of proof assistant: an interactive proof editor, or other interface
User interface

The user interface is the aggregate of means by which people—the User s—Interaction with the system—a particular machine, device, computer program or other complex tools....
, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer
Computer

A computer is a machine that manipulates Data according to a list of Code .The first devices that resemble modern computers date to the mid-20th century , although the computer concept and various machines similar to computers existed earlier....
.

Examples include:






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Encyclopedia


Interactive theorem proving is the field of computer science
Computer science

Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems....
 and mathematical logic
Mathematical logic

Mathematical logic is a subfield of mathematics and logic with close connections to computer science and philosophical logic. The field includes the mathematical study of logic and the applications of formal logic to other areas of mathematics....
 concerned with tools to develop formal proof
Formal proof

A formal proof or derivation is a finite sequence of proposition each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference....
s by man-machine collaboration. This involves some sort of proof assistant: an interactive proof editor, or other interface
User interface

The user interface is the aggregate of means by which people—the User s—Interaction with the system—a particular machine, device, computer program or other complex tools....
, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer
Computer

A computer is a machine that manipulates Data according to a list of Code .The first devices that resemble modern computers date to the mid-20th century , although the computer concept and various machines similar to computers existed earlier....
.

Examples include:
  • HOL theorem prover
    HOL theorem prover

    HOL denotes a family of interactive theorem proving systems sharingsimilar logics and implementation strategies. Systems in this family follow the LCF approach as they are implemented as a library in some programming language....
    s - A family of tools ultimately derived from the LCF theorem prover
    LCF theorem prover

    LCF is an interactive automated theorem prover developed at the universities of University of Edinburgh and Stanford University by Robin Milner and others....
    . 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:
    • - The "primary descendant". Moscow ML
      Moscow ML

      Moscow ML is an implementation of Standard ML.The codebase is derived from Caml Light.The latest release is 2.01. Supported platforms include Unix, Microsoft Windows, Mac OS and .NET Framework....
       based.
    • HOL Lite
      HOL Light theorem prover

      HOL Light is a member of the HOL theorem prover family. Likethe other members, it is a proof assistant for classicalhigher order logic. Compared with other HOL systems, HOL Light is...
       - A thriving "minimalist fork". OCaml based.
    • Isabelle
      Isabelle theorem prover

      The Isabelle theorem prover is an interactive theorem prover framework, a successor of the HOL theorem prover. It is a Logic for Computable Functions LCF theorem prover-style theorem prover , so it is based on a small logical core guaranteeing logical correctness....
       - With a BSD license. Based on Standard ML
      Standard ML

      Standard ML is a general-purpose, Module , functional programming language with compile-time type checking and type inference. It is popular among compiler writers and programming language researchers, as well as in the development of automated theorem proving....
      .
    • ProofPower - Went proprietary, then returned to open source. Based on Standard ML
      Standard ML

      Standard ML is a general-purpose, Module , functional programming language with compile-time type checking and type inference. It is popular among compiler writers and programming language researchers, as well as in the development of automated theorem proving....
      .
  • Prototype Verification System
    Prototype Verification System

    PVS, or the Prototype Verification System, is a specification language integrated with support tools and a theorem prover.It was developed at the Computer Science Laboratory of SRI International, California, USA....
     (PVS) - a proof language and system based on higher-order logic
  • Coq
    Coq

    In computer science, Coq is a proof assistant application. It allows the expression of mathematics 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....
     - Which 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.
  • PhoX
    PhoX

    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 it through subgoals and evidence to prove that goal....
     - A proof assistant based on higher-order logic which is eXtensible
  • MINLOG - A proof assistant based on first-order minimal logic.
  • Matita - A light system based on the Calculus of Inductive Constructions.


See also

  • Automated theorem proving
    Automated theorem proving

    Automated theorem proving or automated deduction, currently the most well-developed subfield of automated reasoning , is the mathematical proof of mathematical theorems by a computer program....
  • Computer-assisted proof
    Computer-assisted proof

    A 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 Proof by exhaustion of a mathematical theorem....
  • Proof verification