Non-Linear Pattern-Matching against Unfree Data Types with Lexical Scoping

This paper proposes a pattern-matching system that enables nonlinear pattern-matching against unfree data types. The system allows multiple occurrences of the same variables in a pattern, multiple results of pattern-matching and modularization of the way of pattern-matching for each data type at the same time. It enables us to represent patternmatching against not only algebraic data types but also unfree data types such as sets, graphs and any other data types whose data have no canonical form and multiple ways of decomposition. I have realized that with a rule that pattern-matching is executed from the left side of a pattern and a rule that a binding to a variable in a pattern can be referred to in its right side of the pattern. Furthermore, I have realized modularization of these patterns with lexical scoping. In my system, a pattern is not a first class object, but a pattern-function that obtains only patterns and returns a pattern is a first class object. This restriction simplifies the non-linear pattern-matching system with lexical scoping. I have already implemented the pattern-matching system in the Egison programming language.