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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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Cited by 14 (5 self)
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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
Verifying mixed realinteger quantifier elimination
 IJCAR 2006, LNCS 4130
, 2006
"... We present a formally verified quantifier elimination procedure for the first order theory over linear mixed realinteger arithmetics in higherorder logic based on a work by Weispfenning. To this end we provide two verified quantifier elimination procedures: for Presburger arithmitics and for lin ..."
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Cited by 8 (5 self)
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We present a formally verified quantifier elimination procedure for the first order theory over linear mixed realinteger arithmetics in higherorder logic based on a work by Weispfenning. To this end we provide two verified quantifier elimination procedures: for Presburger arithmitics and for linear real arithmetics.
A certified verifier for a fragment of Separation logic
 In JSSST Workshop on Programming and Programming Languages (PPL’07). Japan Society for Software Science and Technology
, 2007
"... Separation logic is an extension of Hoare logic to verify imperative programs with pointers and mutable datastructures. Although there exist several implementations of verifiers for separation logic, none of them has actually been itself verified. In this paper, we present a verifier for a fragment ..."
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Cited by 7 (0 self)
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Separation logic is an extension of Hoare logic to verify imperative programs with pointers and mutable datastructures. Although there exist several implementations of verifiers for separation logic, none of them has actually been itself verified. In this paper, we present a verifier for a fragment of separation logic that is verified inside the Coq proof assistant. This verifier is implemented as a Coq tactic by reflection to verify separation logic triples. Thanks to the extraction facility to OCaml, we can also derive a certified, standalone and efficient verifier for separation logic. 1
Proof synthesis and reflection for linear arithmetic. Submitted
, 2006
"... This article presents detailed implementations of quantifier elimination for both integer and real linear arithmetic for theorem provers. The underlying algorithms are those by Cooper (for Z) and by Ferrante and Rackoff (for R). Both algorithms are realized in two entirely different ways: once in ta ..."
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Cited by 6 (5 self)
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This article presents detailed implementations of quantifier elimination for both integer and real linear arithmetic for theorem provers. The underlying algorithms are those by Cooper (for Z) and by Ferrante and Rackoff (for R). Both algorithms are realized in two entirely different ways: once in tactic style, i.e. by a proofproducing functional program, and once by reflection, i.e. by computations inside the logic rather than in the metalanguage. Both formalizations are highly generic because they make only minimal assumptions w.r.t. the underlying logical system and theorem prover. An implementation in Isabelle/HOL shows that the reflective approach is between one and two orders of magnitude faster. 1
Generic proof synthesis for presburger arithmetic
, 2003
"... We develop in complete detail an extension of Cooper’s decision procedure for Presburger arithmetic that returns a proof of the equivalence of the input formula to a quantifierfree formula. For closed input formulae this is a proof of their validity or unsatisfiability. The algorithm is formulated ..."
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Cited by 4 (3 self)
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We develop in complete detail an extension of Cooper’s decision procedure for Presburger arithmetic that returns a proof of the equivalence of the input formula to a quantifierfree formula. For closed input formulae this is a proof of their validity or unsatisfiability. The algorithm is formulated as a functional program that makes only very minimal assumptions w.r.t. the underlying logical system and is therefore easily adaptable to specific theorem provers. 1 Presburger arithmetic Presburger arithmetic is firstorder logic over the integers with + and <. Presburger [3] first showed its decidability. We extend Cooper’s decision procedure [1] such that a successful run returns a proof of the input formula. The atomic PAformulae are defined by Atom:
Mechanized quantifier elimination for linear realarithmetic in Isabelle/HOL
"... Abstract. We integrate Ferrante and Rackoff’s quantifier elimination procedure for linear real arithmetic in Isabelle/HOL in two manners: (a) tacticstyle, i.e. for every problem instance a proof is generated by invoking a series of inference rules, and (b) reflection, where the whole algorithm is i ..."
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Cited by 1 (0 self)
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Abstract. We integrate Ferrante and Rackoff’s quantifier elimination procedure for linear real arithmetic in Isabelle/HOL in two manners: (a) tacticstyle, i.e. for every problem instance a proof is generated by invoking a series of inference rules, and (b) reflection, where the whole algorithm is implemented and verified within Isabelle/HOL. We discuss the performance obtained for both integrations. 1
Reflecting Quantifier Elimination for Linear Arithmetic
"... Abstract. This paper formalizes and verifies quantifier elimination procedures for dense linear orders and for real and integer linear arithmetic in the theorem prover ..."
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Cited by 1 (1 self)
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Abstract. This paper formalizes and verifies quantifier elimination procedures for dense linear orders and for real and integer linear arithmetic in the theorem prover
Parametric linear arithmetic over ordered fields in Isabelle/HOL
"... We use higherorder logic to verify a quantifier elimination procedure for linear arithmetic over ordered fields, where the coefficients of variables are multivariate polynomials over another set of variables, we call parameters. The procedure generalizes Ferrante and Rackoff’s algorithm for the non ..."
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We use higherorder logic to verify a quantifier elimination procedure for linear arithmetic over ordered fields, where the coefficients of variables are multivariate polynomials over another set of variables, we call parameters. The procedure generalizes Ferrante and Rackoff’s algorithm for the nonparametric case. The formalization is based on axiomatic type classes and automatically carries over to e.g. the rational, real and nonstandard real numbers. It is executable, can be applied to HOL formulae by reflection and performs well on practical examples.
A Trustworthy, Extensible Theorem Prover Ph.D. Dissertation Proposal
"... 2.1 Formal verification........................ 3 2.2 Our choice of logic........................ 4 ..."
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2.1 Formal verification........................ 3 2.2 Our choice of logic........................ 4