A survey of the Theorema project

Abstract

The Theorems project aims at extending current computer algebra systems by facilities jor supporting mathematical proving. The present early-prototype version of the Theorems software system is implemented in Mathetnatica 3.0. The system consists of a general higher-order predicate logic prover and a collection of special provers that call each other depending on the particular proof situations. The individual provers imitate the proof style of human mathematicians and aim at producing human-readable proofs in natuml language presented in nested cells that facilitate studying the computer-generated proofs at various levels of detail. The special provers are intimately connected with the junctors that build up the various mathematical domains.1 The Objectives of the Theorems ProjectThe Tlaeorema project aims at providing a uniform (logic and software) frame for computing, solving, and proving. In a simplified view, given a "knowledge base" K of formulae (and a logical / computational derivation mechanism L), q q q a " comput e~' for K (in an abstract sense) enables the user to provide an expression (term, formula, program) T with a free variable z and a value v (from an appropriate domain) and "evaluates" Tr+. (T with rJsubstituted for x) w.r.t. K (within the calculus L), a " solve~' for K enables the user to provide an expression T with a free variable z and produces (all) values v for which Tz+" holds (in L) w.r. t. K, and a "prover" for K enables the user to provide an expression T with a free variable z and decides whether, for all values v, Tz+v holds (in L) w.r.t. K. qThis paper was preparedduring a stay of the first author as a visiting researchfellow at the Universityof Tsukuba, Japan, Chair of ProfessorTetsuo Ida, and was supportedby TARA (Tsukuba Advanced Research Alliance), IPA-AITP (Advanced Information Technology Program of the Information-TechnologyPromotion Agency,

Scheme-Based Systematic Exploration of Natural Numbers

“…We will not, therefore, compare our implementation to all these systems. The provers we wrote in order to carry out this research are based on the existing Theorema natural number induction provers, see [12]. However, our provers allow the user to use any language (s)he chooses, and we implemented more induction rules, such as the complete induction.…”

C-CoRN, the Constructive Coq Repository at Nijmegen

“…The most important of these are: Mizar [28], HOL98 [33] and HOL Light [24], Isabelle/Isar [30], NuPRL/Meta-PRL [12] and PVS [31]. (Other systems for formalization of mathematics, like for instance Theorema [5] and Ωmega [32], do not have large libraries. )…”

Comparing Mathematical Provers

“…During the years the same dream has been at the core of various proof checking projects. Examples are the Automath project [17], the Mizar project [15,23], the NuPRL project [8], and the Theorema project [7]. Recently, the checking of mathematical proofs has become popular in the context of verification of hardware and software: important systems in this area are ACL2 [13] and PVS [18].…”