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An Even Closer Integration of Linear Arithmetic into Inductive Theorem Proving
 Proc. Calculemus 2005: 12 th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning
, 2006
"... To broaden the scope of decision procedures for linear arithmetic, they have to be integrated into theorem provers. Successful approaches e.g. in NQTHM or ACL2 suggest a close integration scheme which augments the decision procedures with lemmas about userdefined operators. We propose an even close ..."
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Cited by 6 (1 self)
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To broaden the scope of decision procedures for linear arithmetic, they have to be integrated into theorem provers. Successful approaches e.g. in NQTHM or ACL2 suggest a close integration scheme which augments the decision procedures with lemmas about userdefined operators. We propose an even closer integration providing feedback about the state of the decision procedure in terms of entailed formulas for three reasons: First, to provide detailed proof objects for proof checking and archiving. Second, to analyze and improve the interaction between the decision procedure and the theorem prover. Third, to investigate whether the communication of the state of a failed proof attempt to the human user with the comprehensible standard GUI mechanisms of the theorem prover can enhance the speculation of auxiliary lemmas.
A comparison of decision procedures
 in Presburger arithmetic. LIRA '97, Univ. of Novi Sad
, 1997
"... The efficient combining and augmenting of decision procedures are often very important for a successful use of theorem provers. There are several schemes for combining and augmenting decision procedures; some of them support handling uninterpreted functions, use of available lemmas, and the like. In ..."
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Cited by 2 (0 self)
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The efficient combining and augmenting of decision procedures are often very important for a successful use of theorem provers. There are several schemes for combining and augmenting decision procedures; some of them support handling uninterpreted functions, use of available lemmas, and the like. In this paper we introduce a general setting for describing different schemes for both combining and augmenting decision procedures. This setting is based on the macro inference rules used in different approaches. Some of these rules are abstraction, entailment, congruence closure and lemma invoking. The general setting gives a simple description and the key ideas of one scheme and makes different schemes comparable. Also, it makes easier combining ideas from different schemes. In this paper we describe several schemes via introduced macro inference rules and report on our prototype implementation.
Strict General Setting for Building Decision Procedures into Theorem Provers
 THE 1ST INTERNATIONAL JOINT CONFERENCE ON AUTOMATED REASONING (IJCAR2001) — SHORT PAPERS
, 2001
"... The efficient and flexible incorporating of decision procedures into theorem provers is very important for their successful use. There are several approaches for combining and augmenting of decision procedures; some of them support handling uninterpreted functions, congruence closure, lemma invoking ..."
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Cited by 2 (1 self)
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The efficient and flexible incorporating of decision procedures into theorem provers is very important for their successful use. There are several approaches for combining and augmenting of decision procedures; some of them support handling uninterpreted functions, congruence closure, lemma invoking etc. In this paper we present a variant of one general setting for building decision procedures into theorem provers (gs framework [18]). That setting is based on macro inference rules motivated by techniques used in different approaches. The general setting enables a simple describing of different combination/augmentation schemes. In this paper, we further develop and extend this setting by an imposed ordering on the macro inference rules. That ordering leads to a ”strict setting”. It makes implementing and using variants of wellknown or new schemes within this framework a very easy task even for a nonexpert user. Also, this setting enables easy comparison of different combination/augmentation schemes and combination of their ideas.
A general setting for . . . decision procedures
, 2002
"... The efficient combining and augmenting of decision procedures are often very important for a successful use of theorem provers. There are several schemes for combining and augmenting decision procedures; some of them support handling uninterpreted functions, use of available lemmas, and the like. In ..."
Abstract
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The efficient combining and augmenting of decision procedures are often very important for a successful use of theorem provers. There are several schemes for combining and augmenting decision procedures; some of them support handling uninterpreted functions, use of available lemmas, and the like. In this paper we introduce a general setting for describing different schemes for both combining and augmenting decision procedures. This setting is based on the macro inference rules used in different approaches. Some of these rules are abstraction, entailment, congruence closure and lemma invoking. The general setting gives a simple description and the key ideas of one scheme and makes different schemes comparable. Also, it makes easier combining ideas from different schemes. In this paper we describe several schemes via introduced macro inference rules and report on our prototype implementation.
Deciding Properties of Lists using Containers
 JOURNAL OF AUTOMATED REASONING
"... We exploit the ability to represent data types as container functors [2,1,3] to develop a novel approach to proving properties of lists using arithmetic decision procedures. Containers capture the idea that concrete data types can be characterised by specifying the shape values take and for every po ..."
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We exploit the ability to represent data types as container functors [2,1,3] to develop a novel approach to proving properties of lists using arithmetic decision procedures. Containers capture the idea that concrete data types can be characterised by specifying the shape values take and for every possible shape, explaining where positions within that shape are stored. More importantly, a representation theorem guarantees that polymorphic functions between container data types are given by container morphisms, which are characterised by mappings between shapes and positions. The key to our approach is to restrict the shape maps of container morphisms to functions that have decidable equality, but which allow for a large class of functions. We also capture the behaviour of position mappings of container morphisms as functions on the natural numbers. The shape maps which we consider are given by piecewiselinear functions, of type N n → N. Such functions are decidable, and this enables us to implement decision procedures for lists.