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On Extensibility of Proof Checkers
 in Dybjer, Nordstrom and Smith (eds), Types for Proofs and Programs: International Workshop TYPES'94, Bastad
, 1995
"... This paper is about mechanical checking of formal mathematics. Given some formal system, we want to construct derivations in that system, or check the correctness of putative derivations; our job is not to ascertain truth (that is the job of the designer of our formal system), but only proof. Howeve ..."
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This paper is about mechanical checking of formal mathematics. Given some formal system, we want to construct derivations in that system, or check the correctness of putative derivations; our job is not to ascertain truth (that is the job of the designer of our formal system), but only proof. However, we are quite rigid about this: only a derivation in our given formal system will do; nothing else counts as evidence! Thus it is not a collection of judgements (provability), or a consequence relation [Avr91] (derivability) we are interested in, but the derivations themselves; the formal system used to present a logic is important. This viewpoint seems forced on us by our intention to actually do formal mathematics. There is still a question, however, revolving around whether we insist on objects that are immediately recognisable as proofs (direct proofs), or will accept some metanotations that only compute to proofs (indirect proofs). For example, we informally refer to previously proved results, lemmas and theorems, without actually inserting the texts of their proofs in our argument. Such an argument could be made into a direct proof by replacing all references to previous results by their direct proofs, so it might be accepted as a kind of indirect proof. In fact, even for very simple formal systems, such an indirect proof may compute to a very much bigger direct proof, and if we will only accept a fully expanded direct proof (in a mechanical proof checker for example), we will not be able to do much mathematics. It is well known that this notion of referring to previous results can be internalized in a logic as a cut rule, or Modus Ponens. In a logic containing a cut rule, proofs containing cuts are considered direct proofs, and can be directly accepted by a proof ch...
Verified Real Number Calculations: A Library for Interval Arithmetic
, 2007
"... Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally est ..."
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Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theoremprover interaction.
Proof Search and Proof Check for Equational and Inductive Theorems
 Conference on Automated Deduction  CADE19
, 2003
"... Abstract. This paper presents ongoing researches on theoretical and practical issues of combining rewriting based automated theorem proving and userguided proof development, with the strong constraint of safe cooperation of both. In practice, we instantiate the theoretical study on the Coq proof a ..."
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Abstract. This paper presents ongoing researches on theoretical and practical issues of combining rewriting based automated theorem proving and userguided proof development, with the strong constraint of safe cooperation of both. In practice, we instantiate the theoretical study on the Coq proof assistant and the ELAN rewriting based system, focusing first on equational and then on inductive proofs. Different concepts, especially rewriting calculus and deduction modulo, contribute to define and to relate proof search, proof representation and proof check.
A reflective extension of ELAN
 Electronic Notes in Theoretical Computer Science
, 1996
"... The expressivity of rewriting logic as metalogic has been already convincingly illustrated. The goal of this paper is to explore the reflective capabilities of ELAN, a language based on the concepts of computational systems and rewriting logic. We define a universal theory for the class of ELAN pro ..."
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The expressivity of rewriting logic as metalogic has been already convincingly illustrated. The goal of this paper is to explore the reflective capabilities of ELAN, a language based on the concepts of computational systems and rewriting logic. We define a universal theory for the class of ELAN programs and the representation function associated to this universal theory. Then we detail the effective transformations to implement and propose the definition of two builtin modules that provide the last step to get the reflective capabilities we want for the ELAN system. 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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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:
Case Studies in MetaLevel Theorem Proving
 PROC. INTL. CONF. ON THEOREM PROVING IN HIGHER ORDER LOGICS (TPHOLS), LECTURE
, 1998
"... We describe an extension of the Pvs system that provides a reasonably efficient and practical notion of reflection and thus allows for soundly adding formalized and verified new proof procedures. These proof procedures work on representations of a part of the underlying logic and their correct ..."
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We describe an extension of the Pvs system that provides a reasonably efficient and practical notion of reflection and thus allows for soundly adding formalized and verified new proof procedures. These proof procedures work on representations of a part of the underlying logic and their correctness is expressed at the object level using a computational reflection function. The implementation of the Pvs system has been extended with an efficient evaluation mechanism, since the practicality of the approach heavily depends on careful engineering of the core system, including efficient normalization of functional expressions. We exemplify the process of applying metalevel proof procedures with a detailed description of the encoding of cancellation in commutative monoids and of the kernel of a BDD package.
Towards Practical Reflection for Formal Mathematics
"... Abstract. We describe a design for a system for mathematical theory exploration that can be extended by implementing new reasoners using the logical input language of the system. Such new reasoners can be applied like the builtin reasoners, and it is possible to reason about them, e.g. proving thei ..."
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Abstract. We describe a design for a system for mathematical theory exploration that can be extended by implementing new reasoners using the logical input language of the system. Such new reasoners can be applied like the builtin reasoners, and it is possible to reason about them, e.g. proving their soundness, within the system. This is achieved in a practical and attractive way by adding reflection, i.e. a representation mechanism for terms and formulae, to the system’s logical language, and some knowledge about these entities to the system’s basic reasoners. The approach has been evaluated using a prototypical implementation called MiniTma. It will be incorporated into the Theorema system. 1
Directly reflective metaprogramming
 Journal of Higher Order and Symbolic Computation
, 2008
"... Existing metaprogramming languages operate on encodings of programs as data. This paper presents a new metaprogramming language, based on an untyped lambda calculus, in which structurally reflective programming is supported directly, without any encoding. The language features callbyvalue and ca ..."
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Existing metaprogramming languages operate on encodings of programs as data. This paper presents a new metaprogramming language, based on an untyped lambda calculus, in which structurally reflective programming is supported directly, without any encoding. The language features callbyvalue and callbyname lambda abstractions, as well as novel reflective features enabling the intensional manipulation of arbitrary program terms. The language is scope safe, in the sense that variables can neither be captured nor escape their scopes. The expressiveness of the language is demonstrated by showing how to implement quotation and evaluation operations, as proposed by Wand. The language’s utility for metaprogramming is further demonstrated through additional representative examples. A prototype implementation is described and evaluated.
Integrating model checking and theorem proving in a reflective functional language
 In IFM
, 2004
"... Abstract. Forte is a formal verification system developed by Intel’s Strategic CAD Labs for applications in hardware design and verification. Forte integrates model checking and theorem proving within a functional programming language, which both serves as an extensible specification language and al ..."
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Abstract. Forte is a formal verification system developed by Intel’s Strategic CAD Labs for applications in hardware design and verification. Forte integrates model checking and theorem proving within a functional programming language, which both serves as an extensible specification language and allows the system to be scripted and customized. The latest version of this language, called reFLect, has quotation and antiquotation constructs that build and decompose expressions in the language itself. This provides combination of patternmatching and reflection features tailored especially for the Forte approach to verification. This short paper is an abstract of an invited presentation given at the International Conference on Integrated Formal Methods in 2004, in which the philosophy and architecture of the Forte system are described and an account is given of the role of reFLect in the system. 1 The Forte Verification Environment Forte [17] is a formal verification environment that has been very effective on
Combining Advanced Formal Hardware Verification Techniques
, 2007
"... To my parents, Henry and Karen Reeber, and my fiancée, Carrie Pankrast, for all their love, guidance, and support. Acknowledgments Most of all, I would like to thank my thesis advisor, Warren Hunt. Warren always has the amazing ability to give me what I need, before I even ask for it. Furthermore, W ..."
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To my parents, Henry and Karen Reeber, and my fiancée, Carrie Pankrast, for all their love, guidance, and support. Acknowledgments Most of all, I would like to thank my thesis advisor, Warren Hunt. Warren always has the amazing ability to give me what I need, before I even ask for it. Furthermore, Warren has been a source of constant encouragement and guidance, without which I never would have started this dissertation, let alone completed it. I would also like to thank the rest of my dissertation committee, Allen Emerson, Steve Keckler, J Moore, and Anna Slobodova, for all the time and energy they spent reviewing my research and for their great feedback both on the dissertation itself and the earlier dissertation proposal. Anna in particular provided me with copious notes that have significantly improved the quality of this dissertation. Thanks also to Sandip Ray, Simha Sethumadhavan, and Jun Sawada for providing excellent feedback on portions of this dissertation. A number of professors at the University of Texas have influenced my work. My