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22
MetiTarski: An Automatic Theorem Prover for RealValued Special Functions
"... Abstract Many theorems involving special functions such as ln, exp and sin can be proved automatically by MetiTarski: a resolution theorem prover modified to call a decision procedure for the theory of real closed fields. Special functions are approximated by upper and lower bounds, which are typica ..."
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Abstract Many theorems involving special functions such as ln, exp and sin can be proved automatically by MetiTarski: a resolution theorem prover modified to call a decision procedure for the theory of real closed fields. Special functions are approximated by upper and lower bounds, which are typically rational functions derived from Taylor or continued fraction expansions. The decision procedure simplifies clauses by deleting literals that are inconsistent with other algebraic facts. MetiTarski simplifies arithmetic expressions by conversion to a recursive representation, followed by flattening of nested quotients. Applications include verifying hybrid and control systems.
Certified sizechange termination
 In Proc. 21st CADE, volume 4603 of LNAI
, 2007
"... Abstract. We develop a formalization of the SizeChange Principle in Isabelle/HOL and use it to construct formally certified termination proofs for recursive functions automatically. 1 ..."
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Abstract. We develop a formalization of the SizeChange Principle in Isabelle/HOL and use it to construct formally certified termination proofs for recursive functions automatically. 1
Fast Reflexive Arithmetic Tactics the linear case and beyond
 in "Types for Proofs and Programs (TYPES’06)", Lecture Notes in Computer Science
, 2006
"... Abstract. When goals fall in decidable logic fragments, users of proofassistants expect automation. However, despite the availability of decision procedures, automation does not come for free. The reason is that decision procedures do not generate proof terms. In this paper, we show how to design ef ..."
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Abstract. When goals fall in decidable logic fragments, users of proofassistants expect automation. However, despite the availability of decision procedures, automation does not come for free. The reason is that decision procedures do not generate proof terms. In this paper, we show how to design efficient and lightweight reflexive tactics for a hierarchy of quantifierfree fragments of integer arithmetics. The tactics can cope with a wide class of linear and nonlinear goals. For each logic fragment, offtheshelf algorithms generate certificates of infeasibility that are then validated by straightforward reflexive checkers proved correct inside the proofassistant. This approach has been prototyped using the Coq proofassistant. Preliminary experiments are promising as the tactics run fast and produce small proof terms. 1
An Efficient Coq Tactic for Deciding Kleene Algebras
, 2009
"... We present a reflexive tactic for deciding the equational theory of Kleene algebras in the Coq proof assistant. This tactic relies on a careful implementation of efficient finite automata algorithms, so that it solves casual equations almost instantaneously. The corresponding decision procedure was ..."
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We present a reflexive tactic for deciding the equational theory of Kleene algebras in the Coq proof assistant. This tactic relies on a careful implementation of efficient finite automata algorithms, so that it solves casual equations almost instantaneously. The corresponding decision procedure was proved correct and complete; correctness is established w.r.t. any model (including binary relations), by formalising Kozen’s initiality theorem.
Modular SMT Proofs for Fast Reflexive Checking inside Coq ⋆
"... Abstract. We present a new methodology for exchanging unsatisfiability proofs between an untrusted SMT solver and a sceptical proof assistant with computation capabilities like Coq. We advocate modular SMT proofs that separate boolean reasoning and theory reasoning; and structure the communication b ..."
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Abstract. We present a new methodology for exchanging unsatisfiability proofs between an untrusted SMT solver and a sceptical proof assistant with computation capabilities like Coq. We advocate modular SMT proofs that separate boolean reasoning and theory reasoning; and structure the communication between theories using NelsonOppen combination scheme. We present the design and implementation of a Coq reflexive verifier that is modular and allows for finetuned theoryspecific verifiers. The current verifier is able to verify proofs for quantifierfree formulae mixing linear arithmetic and uninterpreted functions. Our proof generation scheme benefits from the efficiency of stateoftheart SMT solvers while being independent from a specific SMT solver proof format. Our only requirement for the SMT solver is the ability to extract unsat cores and generate boolean models. In practice, unsat cores are relatively small and their proof is obtained with a modest overhead by our proofproducing prover. We present experiments assessing the feasibility of the approach for benchmarks obtained from the SMT competition. 1
Improving Coq Propositional Reasoning Using a Lazy CNF Conversion Scheme
"... Abstract. In an attempt to improve automation capabilities in the Coq proof assistant, we develop a tactic for the propositional fragment based on the DPLL procedure. Although formulas naturally arising in interactive proofs do not require a stateoftheart SAT solver, the conversion to clausal for ..."
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Abstract. In an attempt to improve automation capabilities in the Coq proof assistant, we develop a tactic for the propositional fragment based on the DPLL procedure. Although formulas naturally arising in interactive proofs do not require a stateoftheart SAT solver, the conversion to clausal form required by DPLL strongly damages the performance of the procedure. In this paper, we present a reflexive DPLL algorithm formalized in Coq which outperforms the existing tactics. It is tightly coupled with a lazy CNF conversion scheme which, unlike Tseitinstyle approaches, does not disrupt the procedure. This conversion relies on a lazy mechanism which requires slight adaptations of the original DPLL. As far as we know, this is the first formal proof of this mechanism and its Coq implementation raises interesting challenges. 1
Context aware calculation and deduction  Ring equalities via Gröbner Bases in Isabelle
 TOWARDS MECHANIZED MATHEMATICAL ASSISTANTS (CALCULEMUS AND MKM 2007), LNAI
, 2007
"... We address some aspects of a proposed system architecture for mathematical assistants, integrating calculations and deductions by common infrastructure within the Isabelle theorem proving environment. Here calculations may refer to arbitrary extralogical mechanisms, operating on the syntactic struc ..."
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Cited by 6 (5 self)
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We address some aspects of a proposed system architecture for mathematical assistants, integrating calculations and deductions by common infrastructure within the Isabelle theorem proving environment. Here calculations may refer to arbitrary extralogical mechanisms, operating on the syntactic structure of logical statements. Deductions are devoid of any computational content, but driven by procedures external to the logic, following to the traditional “LCF system approach”. The latter is extended towards explicit dependency on abstract theory contexts, with separate mechanisms to interpret both logical and extralogical content uniformly. Thus we are able to implement proof methods that operate on abstract theories and a range of particular theory interpretations. Our approach is demonstrated in Isabelle/HOL by a proofprocedure for generic ring equalities via Gröbner Bases.
A tactic for deciding Kleene algebras
 In 1st Coq Workshop. Tech. Univ
, 2009
"... We present a Coq reflexive tactic for deciding equalities or inequalities in Kleene algebras. This tactic is part of a larger project, whose aim is to provide tools for reasoning about binary relations in Coq: binary relations form a Kleene algebra, where the star operation is the reflexive transiti ..."
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Cited by 6 (2 self)
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We present a Coq reflexive tactic for deciding equalities or inequalities in Kleene algebras. This tactic is part of a larger project, whose aim is to provide tools for reasoning about binary relations in Coq: binary relations form a Kleene algebra, where the star operation is the reflexive transitive closure. Our tactic relies on an initiality theorem, whose proof goes by replaying finite automata algorithms in an algebraic way, using matrices.
A Reflexive Formalization of a SAT Solver in Coq
 In Proceedings of TPHOLs
, 2008
"... Abstract. We present a Coq formalization of an algorithm deciding the satisfiability of propositional formulas (SAT). This SAT solver is described as a set of inference rules in a manner that is independent of the actual representation of propositional variables and formulas. We prove soundness and ..."
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Abstract. We present a Coq formalization of an algorithm deciding the satisfiability of propositional formulas (SAT). This SAT solver is described as a set of inference rules in a manner that is independent of the actual representation of propositional variables and formulas. We prove soundness and completeness for this system, and instantiate our solver directly on the propositional fragment of Coq’s logic in order to obtain a fully reflexive tactic. Such a tactic represents a first and important step towards our ultimate goal of embedding an automated theorem prover inside the Coq system. We also extract a certified Ocaml implementation of the algorithm. 1
Programming and certifying the CAD algorithm inside the coq system
 Mathematics, Algorithms, Proofs, volume 05021 of Dagstuhl Seminar Proceedings, Schloss Dagstuhl
, 2005
"... Abstract. A. Tarski has shown in 1975 that one can perform quantifier elimination in the theory of real closed fields. The introduction of the Cylindrical Algebraic Decomposition (CAD) method has later allowed to design rather feasible algorithms. Our aim is to program a reflectional decision proced ..."
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Abstract. A. Tarski has shown in 1975 that one can perform quantifier elimination in the theory of real closed fields. The introduction of the Cylindrical Algebraic Decomposition (CAD) method has later allowed to design rather feasible algorithms. Our aim is to program a reflectional decision procedure for the Coq system, using the CAD, to decide whether a (possibly multivariate) system of polynomial inequalities with rational coefficients has a solution or not. We have therefore implemented various computer algebra tools like gcd computations, subresultant polynomial or Bernstein polynomials.