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Linear quantifier elimination
 In Automated reasoning (IJCAR), volume 5195 of LNCS
, 2008
"... Abstract. This paper presents verified quantifier elimination procedures for dense linear orders (DLO), for real and for integer linear arithmetic. The DLO procedures are new. All procedures are defined and verified in the theorem prover Isabelle/HOL, are executable and can be applied to HOL formula ..."
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Abstract. This paper presents verified quantifier elimination procedures for dense linear orders (DLO), for real and for integer linear arithmetic. The DLO procedures are new. All procedures are defined and verified in the theorem prover Isabelle/HOL, are executable and can be applied to HOL formulae themselves (by reflection). 1
Proof Synthesis and Reflection for Linear Arithmetic
 J. OF AUT. REASONING
"... 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 t ..."
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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 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.
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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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.
Proof reconstruction for firstorder logic and settheoretical constructions
 Sixth International Workshop on Automated Verification of Critical Systems (AVOCS ’06) – Preliminary Proceedings
, 2006
"... Proof reconstruction is a technique that combines an interactive theorem prover and an automatic one in a sound way, so that users benefit from the expressiveness of the first tool and the automation of the latter. We present an implementation of proof reconstruction for firstorder logic and setth ..."
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Proof reconstruction is a technique that combines an interactive theorem prover and an automatic one in a sound way, so that users benefit from the expressiveness of the first tool and the automation of the latter. We present an implementation of proof reconstruction for firstorder logic and settheoretical constructions between the interactive theorem prover Isabelle and the automatic SMT prover haRVey. 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.
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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Abstract. This paper formalizes and verifies quantifier elimination procedures for dense linear orders and for real and integer linear arithmetic in the theorem prover
Shallow Dependency Pairs
"... Abstract. We show how the dependency pair approach, commonly used to modularize termination proofs of rewrite systems, can be adapted to establish termination of recursive functions in a system like Isabelle/HOL or Coq. It turns out that all that is required are two simple lemmas about wellfoundedne ..."
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Abstract. We show how the dependency pair approach, commonly used to modularize termination proofs of rewrite systems, can be adapted to establish termination of recursive functions in a system like Isabelle/HOL or Coq. It turns out that all that is required are two simple lemmas about wellfoundedness. 1
Formalization of Real Analysis: A Survey of Proof . . .
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
"... In the recent years, numerous proof systems have improved enough to be used for formally verifying nontrivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on proper ..."
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In the recent years, numerous proof systems have improved enough to be used for formally verifying nontrivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability, and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPowerHOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, CCoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis.