Two Computer-Supported Proofs in Metric Space Topology
Every mathematician will agree that the discovery, analysis, and communication of theorems and proofs is at the heart of his or her discipline. A number of computer programs (such as Maple or Mathematica) assist mathematicians in testing conjectures and proving certain kinds of theorems, typically identities involving rational functions or trigonometric polynomials. These systems, however, were never intended for some of the most critical parts of the mathematical process: formulation of concepts and theories and rigorous proof of general theorems. There are other programs, less well known in the mathematical community, that are designed to provide computer support for the actual theorem proving process. These programs, usually called theorem provers, tend to be very specialized tools, aimed at discovering or checking proofs in languages or logical systems not ordinarily used by mathematicians. Nevertheless, some theorem provers have been used to produce fully machine-checked proofs of mathematically significant results; for examples, see [1, 2, 3, 8, 9]. In this article we discuss two proofs that were created with the help of a computer theorem proving system called imps (Interactive Mathematical Proof System), which is currently being developed at The MITRE Corporation. The fact that these results were proven using a computer is not ∗Supported by the MITRE-Sponsored Research Program. Published in the Computers and Mathematics column of Notices of the American Mathematical Society 38 (1991) 1133–1138.