### Unification of Higher-order Patterns modulo Simple Syntactic Equational Theories

*Alexandre Boudet*

#### Abstract

We present an algorithm for unification of higher-order patterns modulo simple syntactic equational theories as defined by Kirchner [14]. The algorithm by Miller [17] for pattern unification, refined by Nipkow [18] is first modified in order to behave as a first-order unification algorithm. Then the mutation rule for syntactic theories of Kirchner [13,14] is adapted to pattern E-unification. If the syntactic algorithm for a theory E terminates in the first-order case, then our algorithm will also terminate for pattern E-unification. The result is a DAG-solved form plus some equations of the form λx.F(x) = λ x. F(xπ) where xπ is a permutation of x When all function symbols are decomposable these latter equations can be discarded, otherwise the compatibility of such equations with the solved form remains open.

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