This article describes a general framework for designing purely functional datatypes that automatically satisfy given size or structural constraints. Using the framework we develop implementations of different matrix types (for example, square matrices) and implementations of several tree types (for example, Braun trees and 2-3 trees). Consider representing square n×n matrices. The usual representation using lists of lists fails to meet the structural constraints: there is no way to ensure that the outer list and the inner lists have the same length. The main idea of our approach is to solve in a first step a related, but simpler problem, namely to generate the multiset of all square numbers. To describe this multiset we employ recursion equations involving finite multisets, multiset union, addition and multiplication lifted to multisets. In a second step we mechanically derive from these recursion equations datatype definitions that enforce the ‘squareness’ constraint. The transformation makes essential use of parameterized types.
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