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Regaining Cut Admissibility in Deduction Modulo using Abstract Completion
, 2009
"... Deduction modulo is a way to combine computation and deduction in proofs, by applying the inference rules of a deductive system (e.g. natural deduction or sequent calculus) modulo some congruence that we assume here to be presented by a set of rewrite rules. Using deduction modulo is equivalent to p ..."
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Deduction modulo is a way to combine computation and deduction in proofs, by applying the inference rules of a deductive system (e.g. natural deduction or sequent calculus) modulo some congruence that we assume here to be presented by a set of rewrite rules. Using deduction modulo is equivalent to proving in a theory corresponding to the rewrite rules, and leads to proofs that are often shorter and more readable. However, cuts may be not admissible anymore. We define a new system, the unfolding sequent calculus, and prove its equivalence with the sequent calculus modulo, especially w.r.t. cutfree proofs. It permits to show that it is even undecidable to know if cuts can be eliminated in the sequent calculus modulo a given rewrite system. Then, to recover the cut admissibility, we propose a procedure to complete the rewrite system such that the sequent calculus modulo the resulting system admits cuts. This is done by generalizing the KnuthBendix completion in a nontrivial way, using the framework of abstract canonical systems. These
A CurryHowardDe Bruijn Isomorphism Modulo. Under submission
, 2006
"... The rewriting calculus combines in a unified setting the frameworks and capabilities of rewriting and calculus. Its most general typed version, called Pure Pattern Type Systems (P 2TS) and adapted from Barendregt’s cube, is especially interesting from a logical point of view. We show how to use a ..."
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The rewriting calculus combines in a unified setting the frameworks and capabilities of rewriting and calculus. Its most general typed version, called Pure Pattern Type Systems (P 2TS) and adapted from Barendregt’s cube, is especially interesting from a logical point of view. We show how to use a subset of P 2TS as a proofterm language for natural deduction modulo, extending the CurryHowardDe Bruijn isomorphism for this class of logical formalisms. The pattern matching featured in the calculus allows us to model any congruence given by a term rewriting system. We characterize how proofs can be denoted by P 2TS terms and we discuss the interest of our proofterm language for the issue of cut elimination. Finally, we explore some relations between our proofterm language and other formalisms: extraction of terms and/or rewrite rules from P 2TSterms, but also automated generation of proofterms by a rewritingbased language. 1
Mechanized quantifier elimination for linear realarithmetic in Isabelle/HOL
"... 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 ..."
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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.
Type Theory with FirstOrder Data Types and SizeChange Termination
, 2004
"... We prove normalization for a dependently typed lambdacalculus extended with firstorder data types and computation schemata for firstorder sizechange terminating recursive functions. Sizechange termination, introduced by C.S. Lee, N.D. Jones and A.M. BenAmram, can be seen as a generalized form ..."
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We prove normalization for a dependently typed lambdacalculus extended with firstorder data types and computation schemata for firstorder sizechange terminating recursive functions. Sizechange termination, introduced by C.S. Lee, N.D. Jones and A.M. BenAmram, can be seen as a generalized form of structural induction, which allows inductive computations and proofs to be defined in a straightforward manner. The language can be used as a proof system—an extension of MartinLöf’s Logical Framework.
Checking foundational proof certificates for firstorder logic
"... We present the design philosophy of a proof checker based on a notion of foundational proof certificates. This checker provides a semantics of proof evidence using recent advances in the theory of proofs for classical and intuitionistic logic. That semantics is then performed by a (higherorder) log ..."
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We present the design philosophy of a proof checker based on a notion of foundational proof certificates. This checker provides a semantics of proof evidence using recent advances in the theory of proofs for classical and intuitionistic logic. That semantics is then performed by a (higherorder) logic program: successful performance means that a formal proof of a theorem has been found. We describe how the λProlog programming language provides several features that help guarantee such a soundness claim. Some of these features (such as strong typing, abstract datatypes, and higherorder programming) were features of the ML programming language when it was first proposed as a proof checker for LCF. Other features of λProlog (such as support for bindings, substitution, and backtracking search) turn out to be equally important for describing and checking the proof evidence encoded in proof certificates. Since trusting our proof checker requires trusting a programming language implementation, we discuss various avenues for enhancing one’s trust of such a checker. 1
Proof Certification in Zenon Modulo: When Achilles Uses Deduction Modulo to Outrun the Tortoise with Shorter Steps
 in "IWIL  10th International Workshop on the Implementation of Logics  2013
, 2013
"... We present the certifying part of the Zenon Modulo automated theorem prover, which is an extension of the Zenon tableaubased first order automated theorem prover to deduction modulo. The theory of deduction modulo is an extension of predicate calculus, which allows us to rewrite terms as well as pr ..."
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We present the certifying part of the Zenon Modulo automated theorem prover, which is an extension of the Zenon tableaubased first order automated theorem prover to deduction modulo. The theory of deduction modulo is an extension of predicate calculus, which allows us to rewrite terms as well as propositions, and which is well suited for proof search in axiomatic theories, as it turns axioms into rewrite rules. In addition, deduction modulo allows Zenon Modulo to compress proofs by making computations implicit in proofs. To certify these proofs, we use Dedukti, an external proof checker for the λΠcalculus modulo, which can deal natively with proofs in deduction modulo. To assess our approach, we rely on some experimental results obtained on the benchmarks provided by the TPTP library. 1
An Open Logical Framework
"... The LFP Framework is an extension of the HarperHonsellPlotkin’s Edinburgh Logical Framework LF with external predicates, hence the name Open Logical Framework. This is accomplished by defining lock type constructors, which are a sort of ⋄modality constructors, releasing their argument under the ..."
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The LFP Framework is an extension of the HarperHonsellPlotkin’s Edinburgh Logical Framework LF with external predicates, hence the name Open Logical Framework. This is accomplished by defining lock type constructors, which are a sort of ⋄modality constructors, releasing their argument under the condition that a possibly external predicate is satisfied on an appropriate typed judgement. Lock types are defined using the standard pattern of constructive type theory, i.e. via introduction, elimination, and equality rules. Using LFP, one can factor out the complexity of encoding specific features of logical systems which would otherwise be awkwardly encoded in LF, e.g. sideconditions in the application of rules in Modal Logics, and substructural rules, as in noncommutative Linear Logic. The idea of LFP is that these conditions need only to be specified, while their verification can be delegated to an external proof engine, in the style of the Poincaré Principle or Deduction Modulo. Indeed such paradigms can be adequately formalized in LFP. We investigate and characterize the metatheoretical properties of the calculus underpinning LFP: strong normalization, confluence, and subject reduction. This latter property holds under the assumption that the predicates are wellbehaved, i.e. closed under weakening, permutation, substitution, and reduction in the arguments. Moreover, we
Theorem Proving Modulo Based on Boolean Equational Procedures
, 2007
"... The moral of my story is that if we treat our formalisms with the care and respect that we pay to our other subtle artifacts, our care and respect will be more than rewarded. Calculemus! E.W. Dijkstra, “How computer science created a new mathematical style” EWD1073. II Table of Contents ..."
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The moral of my story is that if we treat our formalisms with the care and respect that we pay to our other subtle artifacts, our care and respect will be more than rewarded. Calculemus! E.W. Dijkstra, “How computer science created a new mathematical style” EWD1073. II Table of Contents
Systems for Integrated . . .  Interim Report of the CALCULEMUS Network.
"... This document reports on the research progress made in all work task of the CALCULEMUS IHP Training Network HPRNCT200000102 after the first half of the 48 months funding period. The objectives of the CALCULEMUS Network are: 1. outline the design of a new generation of mathematical software system ..."
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This document reports on the research progress made in all work task of the CALCULEMUS IHP Training Network HPRNCT200000102 after the first half of the 48 months funding period. The objectives of the CALCULEMUS Network are: 1. outline the design of a new generation of mathematical software systems and computeraided verification tools; 2. the training of young researchers in the broad field of mechanical reasoning and formal methods; 3. the dissemination of the results both in industry and in academia; and 4. the crossfertilisation and amalgamation of the automated theorem proving (ATP/DS), computer algebra (CAS), term rewriting systems (TRS) interactive proof development systems (ITP) and software