### Abstract

We formalize first-order query evaluation over an infinite domain with
equality. We first define the syntax and semantics of first-order
logic with equality. Next we define a locale

We provide an interpretation of the locale

*eval_fo*abstracting a representation of a potentially infinite set of tuples satisfying a first-order query over finite relations. Inside the locale, we define a function*eval*checking if the set of tuples satisfying a first-order query over a database (an interpretation of the query's predicates) is finite (i.e., deciding*relative safety*) and computing the set of satisfying tuples if it is finite. Altogether the function*eval*solves*capturability*(Avron and Hirshfeld, 1991) of first-order logic with equality. We also use the function*eval*to prove a code equation for the semantics of first-order logic, i.e., the function checking if a first-order query over a database is satisfied by a variable assignment.We provide an interpretation of the locale

*eval_fo*based on the approach by Ailamazyan et al. A core notion in the interpretation is the active domain of a query and a database that contains all domain elements that occur in the database or interpret the query's constants. We prove the main theorem of Ailamazyan et al. relating the satisfaction of a first-order query over an infinite domain to the satisfaction of this query over a finite domain consisting of the active domain and a few additional domain elements (outside the active domain) whose number only depends on the query. In our interpretation of the locale*eval_fo*, we use a potentially higher number of the additional domain elements, but their number still only depends on the query and thus has no effect on the data complexity (Vardi, 1982) of query evaluation. Our interpretation yields an*executable*function*eval*. The time complexity of*eval*on a query is linear in the total number of tuples in the intermediate relations for the subqueries. Specifically, we build a database index to evaluate a conjunction. We also optimize the case of a negated subquery in a conjunction. Finally, we export code for the infinite domain of natural numbers.