Abstract. We present a fully proof-producing implementation of a quantifierelimination procedure for real closed fields. To our knowledge, this is the first generally useful proof-producing implementation of such an algorithm. Whilemany problems within the domain are intractable, we demonstrate convincing examples of its value in interactive theorem proving. 1 Overview and related work Arguably the first automated theorem prover ever written was for a theory of lineararithmetic [8]. Nowadays many theorem proving systems, even those normally classified as `interactive ' rather than `automatic', contain procedures to automate routinearithmetical reasoning over some of the supported number systems like N, Z, Q, R and C. Experience shows that such automated support is invaluable in relieving users ofwhat would otherwise be tedious low-level proofs. We can identify several very common limitations of such procedures:- Often they are restricted to proving purely universal formulas rather than dealingwith arbitrary quantifier structure and performing general quantifier elimination.- Often they are not complete even for the supported class of formulas; in partic-ular procedures for the integers often fail on problems that depend inherently on divisibility properties (e.g. 8x y 2 Z. 2x + 1 6 = 2y)- They seldom handle non-trivial nonlinear reasoning, even in such simple cases as 8x y 2 R. x> 0 ^ y> 0) xy> 0, and those that do [18] tend to use heuristicsrather than systematic complete methods.- Many of the procedures are standalone decision algorithms that produce no certifi-cate of correctness and do not produce a `proof ' in the usual sense. The earliest serious exception is described in [4]. Many of these restrictions are not so important in practice, since subproblems aris-ing in interactive proof can still often be handled effectively. Indeed, sometimes the restrictions are unavoidable: Tarski's theorem on the undefinability of truth implies thatthere cannot even be a complete semidecision procedure for nonlinear reasoning over

primitive notions of quality and relation. With the introduction of a unary relation, we develop a system totally based on the sole primitive notion of relation. Such a modification enables a definition of the concept of dynamic unary relation. In this way we construct a simple language capable to express other well known theories such as Robinson’s arithmetic or a piece of a theory of concatenation. A key role in this system plays an abstract relation designated by “ ()”, which can be interpreted in different ways, but in this paper we will focus on the case when we can perform computations and obtain results. Keywords—language, unary relations, arithmetic, computability I.