A formalisation of nominal α-equivalence with A, C, and AC function symbols

Abstract This paper describes a formalisation in Coq of nominal syntax extended with associative (A), commutative (C) and associative-commutative (AC) operators. This formalisation is based on a natural notion of nominal α-equivalence, avoiding the use of an auxiliary weak α-relation used in previous formalisations of nominal AC equivalence. A general α-relation between terms with A, C and AC function symbols is specified and formally proved to be an equivalence relation. As corollaries, one obtains the soundness of α-equivalence modulo A, C and AC operators. General α-equivalence problems with A operators are log-linearly bounded in time while if there are also C operators they can be solved in O ( n 2 log ⁡ n ) ; nominal α-equivalence problems that also include AC operators can be solved with the same running time complexity as in standard first-order AC approaches. This development is a first step towards verification of nominal matching, unification and narrowing algorithms modulo equational theories in general.