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Methods for deciding quantifier-free non-linear arithmetical conjectures over R are crucial in the formal verification of many real-world systems and in formalised mathematics. While non-linear (rational function) arithmetic over R is decidable, it is fundamentally infeasible: any general decision method for this problem is worst-case exponential in the… (More)

MetiTarski [1] is an automatic theorem prover that can prove inequalities involving sin, cos, exp, ln, etc. During its proof search, it generates a series of subproblems in nonlinear polynomial real arithmetic which are reduced to true or false using a decision procedure for the theory of real closed fields (RCF). These calls are often a bottleneck: RCF is… (More)

High-performance SMT solvers contain many tightly integrated , hand-crafted heuristic combinations of algorithmic proof methods. While these heuristic combinations tend to be highly tuned for known classes of problems, they may easily perform badly on classes of problems not anticipated by solver developers. This issue is becoming increasingly pressing as… (More)

—Hybrid systems with both discrete and continuous dynamics are an important model for real-world physical systems. The key challenge is how to ensure their correct functioning w.r.t. safety requirements. Promising techniques to ensure safety seem to be model-driven engineering to develop hybrid systems in a well-defined and traceable manner and formal… (More)

Hybrid systems with both discrete and continuous dynamics are an important model for real-world cyber-physical systems. The key challenge is to ensure their correct functioning w.r.t. safety requirements. Promising techniques to ensure safety seem to be model-driven engineering to develop hybrid systems in a well-defined and traceable manner, and formal… (More)

- Leonardo de Moura, Grant Olney Passmore
- SMT '09
- 2009

Hilbert's weak Nullstellensatz guarantees the existence of algebraic proof objects certifying the unsatisfiability of systems of polynomial equations not satisfiable over any algebraically closed field. Such proof objects take the form of ideal membership identities and can be found algorithmically using Gröbner bases and cofactor-based linear algebra… (More)

- Grant Olney Passmore, Leonardo De Moura
- 2009

Using the machinery of proof orders originally introduced by Bach-mair and Dershowitz in the context of canonical equational proofs, we give an abstract, strategy-independent presentation of Gröbner basis procedures and prove the correctness of two classical criteria for recog-nising superfluous S-polynomials, Buchberger's criteria 1 and 2, w.r.t. arbitrary… (More)

Though decidable, the theory of real closed fields (RCF) is fundamentally infeasible. This is unfortunate, as automatic proof methods for nonlinear real arithmetic are crucially needed in both formalised mathematics and the verification of real-world cyber-physical systems. Consequently, many researchers have proposed fast, sound but incomplete RCF proof… (More)

We present a complete, certificate-based decision procedure for first-order univariate polynomial problems in Isabelle [17]. It is built around an executable function to decide the sign of a univariate polynomial at a real algebraic point. The procedure relies on no trusted code except for Isabelle's kernel and code generation. This work is the first step… (More)