Monitoring and Verifying Mathematical Proofs Formulated in a

A restricted natural language is presented which is suitable for formulating mathematical proofs in the domain of calculus. A line of a proof according to the language consists of three parts: A marking, a proof statement, and a foundation of the statement. Foundations include among others the name of a theorem, the name of a concept, or a formula manipulation operation. It is demonstrated how mathematical proofs worded in that language may be automatically monitored and checked for correctness and completeness by a computer program. For that, techniques of the fields of theorem proving and of formula manipulation are applied; the lines of the original proof are transformed into a quantifier free form and checked line by line; an internal knowledge base of concepts and theorems allows for verifying proof statements which are founded by concept definitions or theorem applications. The described methods may be used in virtual or face-to-face universities for the purpose of proof exercises by students or for the purpose of automatically checking and scoring student proofs. The approach together with a medium-grained XML representation of concepts, theorems, and proofs may form the core of a learning environment which gives students the opportunity of an intensive interactive occupation with mathematical proofs.