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An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right. In 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.
An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right. Coq 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.
The Isabelle [a] automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala.As a Logic for Computable Functions (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.
Another approach is deductive verification. [5] [6] It consists of generating from the system and its specifications (and possibly other annotations) a collection of mathematical proof obligations, the truth of which imply conformance of the system to its specification, and discharging these obligations using either proof assistants (interactive theorem provers) (such as HOL, ACL2, Isabelle ...
However, since the tool to find the proof (theorem prover) is implemented in software and is complex, there is a high probability it will contain errors. One approach has been to use a tool that verifies the proof (a proof checker ) which, because it is much simpler than a proof-finder, is less likely to contain errors.
[2]: 60 When the user adds and removes the proof steps, the proof tree is constructed which Jape can show either in a tree shape or in box forms. [5] Jape allows to display proofs at different levels of abstraction. It is also possible to present a forward proof in a natural deduction style by using the specialized modes of display for proofs. [6]
Metamath is a formal language and an associated computer program (a proof assistant) for archiving and verifying mathematical proofs. [2] 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.
A more powerful version, called PC-Nqthm (Proof-checker Nqthm) was developed by Matt Kaufmann. This gave the proof tools that the system uses automatically to the user, so that more guidance can be given to the proof. This is a great help, as the system has an unproductive tendency to wander down infinite chains of inductive proofs.