Theorema: Towards computer-aided mathematical theory exploration

Abstract

Theorema is a project that aims at supporting the entire process of mathematical theory exploration within one coherent logic and software system. This survey paper illustrates the style of Theoremasupported mathematical theory exploration by a case study (the automated synthesis of an algorithm for the construction of Gröbner Bases) and gives an overview on some reasoners and organizational tools for theory exploration developed in the Theorema project.

An Automated Confluence Proof for an Infinite Rewrite System Parametrized over an Integro-Differential Algebra

“…Theorema is a system designed as an integrated environment for doing mathematics, in particular proving, solving, and computing in various domains of mathematics [10]. Implemented on top of the computer algebra system Mathematica, its core language is higher-order predicate logic and contains a natural programming language such that algorithms can be coded and verified in a unified formal frame.…”

Translating between Language and Logic: What Is Easy and What Is Difficult

“…It has motivated systems like STUDENT [2], Mathematical Vernacular [3], Mizar [4], OMEGA [5], Isar [6], Vip [7], Theorema [8], MathLang [9], Naproche [10], and FMathL [11]. These systems permit user interaction in a notation that resembles English more than logical symbolisms do.…”

Automatic Verification of Regular Constructions in Dynamic Geometry Systems

“…The prover can be used in conjunction with other dynamic geometry tools. Apart from the original implementation by its authors [4,5], we are aware of another two geometry provers based on the area method: one within the Theorema project [3], and one within the system Coq (COQareaMethod) [18].…”