Computer Algebra Systems as a Tool for Mathematical Proofs in Quantum Logics

In the last years, computers became a useful tool in some mathematical proofs. We argue that modern computer algebra systems simplify this work. Their usefulness can be demonstrated on recent results in quantum logic theory. We give two examples of highly nontrivial mathematical results which required computer veriication that was performed by CASs. 1 The structure of quantum logic Looking for the appropriate algebraic structure describing the events of a quantum mechanical system, Birkhoo and von Neumann 1] suggested the notion of orthomodular lattice. This structure possesses a common generalization of Boolean algebras and lattices of closed subspaces of Hilbert spaces. We refer to 5] for the basic information about orthomodular lattices. Let us rst review basic notions as we shall use them in the sequel. An orthomodular lattice is a lattice L with a least and a greatest element (denoted by 0, 1, respectively) and with a unary operation 0 (orthocomplemen-tation) satisfying the following properties for all a; b 2 L: 1. (a 0) 0 = a, 2. a b =) b 0 a 0 , 3. a _ a 0 = 1, 4. a b =) b = a _ (b ^ a 0). The authors gratefully acknowledge the support of the project no. VS96049 of the Czech Ministry of Education.