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...midecision procedure for nonlinear reasoning oversthe integers. At the other end of the tower of number systems, one of the few implementations that has none of the above restrictions is described in =-=[16]-=-, but that is for the complex numbers where quantifier elimination is particularly easy. Over the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary...

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...l numbers and/or the development of elementary real analysis. There has also been some work in formalizing complex analysis, with proofs of the Fundamental Theorem of Algebra in Mizar [12], HOL Light =-=[7]-=- and — constructively — in Coq [5]. However, as far as we are aware the first serious formalization of arbitrary N-dimensional Euclidean space is quite recent, undertaken by Hales in HOL Light as part...

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...} The theory of complex numbers is then built on top, using complex (C) as a mere abbreviation for real^2 (R2). Most of the very basic setup was taken over with trivial changes from an earlier theory =-=[14]-=- that was constructed more explicitly Formalizing an Analytic Proof of the PNT 249 from pairs of reals. After being transferred to the new formulation, this theory was slightly revised and greatly ext...

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...7] was the first to implement a proof-producing version of Cooper’s algorithm in a theorem prover. Similar implementation of QE for complex numbers and for real closed fields are reported by Harrison =-=[10]-=- and McLaughlin [14]. The CAD QE procedure for real closed fields has been reflected but only partly verified by Mahboubi [13] in Coq. Acknowledgment Amine Chaieb alerted me to the infinitesimal appro...

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... where the computer algebra system comes into play. Integrating Buchberger algorithm in INRIAsPrimality Proving with Elliptic Curves 7 a safe way inside a prover is possible and has already been done =-=[6, 13]-=-, but cases 1 ÷ 3 can be decided by a much simpler strategy. To illustrate this strategy, let us consider a more elementary example, i.e. the proof that the addition is internal in the tangent case. I...

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...hms and the corresponding floating- and fixed-point implementations as predicates in higher-order logic. For this, we make use of existing theories in HOL on the construction of real [15] and complex =-=[18]-=- numbers, the formalization of IEEE-754 standard based floating-point arithmetic [16, 17], and the formalization of fixed-point arithmetic [1, 2]. We use valuation functions to find the real values of...

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...but this was not our purpose. Efficient decision procedures for the theory of real closed fields usually best deal with the universal fragment of the theory using Gröbner bases computations. Harrison =-=[10]-=- has formalized both quantifier elimination for the theory of complex numbers, and a proof producing version of Buchberger algorithm [6]. An efficient Gröbner based tactic is also available in Coq [15...