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Automating elementary numbertheoretic proofs using Gröbner bases
"... Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates ..."
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Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates such as ‘divisible by’, ‘congruent ’ and ‘coprime’; one notable example in this class is the Chinese Remainder Theorem (for a specific number of moduli). The method is based on a reduction to ideal membership assertions that are then solved using Gröbner bases. As well as illustrating the usefulness of the procedure on examples, and considering some extensions, we prove a limited form of completeness for properties that hold in all rings. 1
Sequents, Frames, and Completeness
"... . Entailment relations, originated from Scott, have been used for describing mathematical concepts constructively and for representing categories of domains. This paper gives an analysis of the freely generated frames from entailment relations. This way, we obtain completeness results under the ..."
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. Entailment relations, originated from Scott, have been used for describing mathematical concepts constructively and for representing categories of domains. This paper gives an analysis of the freely generated frames from entailment relations. This way, we obtain completeness results under the unifying principle of the spatiality of coherence logic. In particular, the domain of disjunctive states, derived from the hyperresolution rule as used in disjunctive logic programs, can be seen as the frame freely generated from the opposite of a sequent structure. At the categorical level, we present equivalences among the categories of sequent structures, distributive lattices, and spectral locales using appropriate morphisms. Key words: sequent structures, lattices, frames, domain theory, resolution, category. Introduction Entailment relations were introduced by Scott as an abstract description of Gentzen's sequent calculus [1315]. It can be seen as a generalisation of the ear...
A Syntactical Proof Of The Marriage Lemma
"... . We give a proof of the classical Marriage Lemma [4] using completeness of hyperresolution. This argument is purely syntactical, and extends directly to the infinite case. As an application we give a purely syntactical version of a proof that resolution is exponential on the pigeonhole principl ..."
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. We give a proof of the classical Marriage Lemma [4] using completeness of hyperresolution. This argument is purely syntactical, and extends directly to the infinite case. As an application we give a purely syntactical version of a proof that resolution is exponential on the pigeonhole principle. Introduction The resolution rule [7] can be stated without references to logical connectives. It manipulates clauses, that can be seen abstractly as a pair of finite sets of "token" or atomic propositions. Despite, or maybe because of, this simplicity, it has deep connections with various parts of mathematics. Some of these connections were pointed out early on in [6, 5], respectively in the field of discrete mathematics and algebra. In [2, 3] other connections are described in the framework of entailment relations [8], an abstract version of Gentzen multiconclusion sequent calculus, which can be seen as a variation on the resolution calculus. In resolution calculus, we are interested ...