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When for a theory E there exists a h i t e presentationA such that any equational theorem can be proven using at most one application of an equational axiom of A at the top occurrence, the theory is called syntactic. We characterize syntactic theories with respect to the unifiability properties of general equations of the form VI,. . . , Vn) = ? g (~n +~, . . . ,v,). This leads to a better understanding of the relationship between syntacticness and unifiability and allows to give a very simple proof of the fact that theories like associativity or associativity and commutativity are syntactic.

Abstract. The Available Bit Rate protocol (ABR) for ATM networks is well-adapted to data traffic by providing minimum rate guarantees and low cell loss to the ABR source end system. An ABR conformance algorithm for controlling the source rates through an interface has been defined by ATM Forum and a more efficient version of it has been designed in [13]. We present in this work the first complete mechanical verification of the equivalence between these two algorithms. The proof is involved and has been supported by the PVS theorem-prover. It has required many lemmas, case analysis and induction reasoning for the manipulation of unbounded scheduling lists. Some ABR conformance protocols have been verified in previous works. However these protocols are approximations of the one we consider here. For instance, the algorithms mechanically proved in [10] and [5] consider scheduling lists with only two elements.

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