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A logical approach to abstract algebra
 Math. Structures Comput. Sci
"... Abstract. Recent work in constructive mathematics show that Hilbert’s program works for a large part of abstract algebra. Using in an essential way the ideas contained in the classical arguments, we can transform a large number of abstract non effective proofs of “concrete ” statements into elementa ..."
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Abstract. Recent work in constructive mathematics show that Hilbert’s program works for a large part of abstract algebra. Using in an essential way the ideas contained in the classical arguments, we can transform a large number of abstract non effective proofs of “concrete ” statements into elementary proofs. Surprisingly the arguments we get are not only elementary but also mathematically clearer and not necessarily longer. We present an example where the simplification was significant enough to suggest an improved version of a classical theorem.
DIFFERENTIAL EQUATIONS, SPENCER COHOMOLOGY, AND COMPUTING RESOLUTIONS
"... Abstract. We propose a new point of view of the Spencer cohomology appearing in the formal theory of differential equations based on a dual approach via comodules. It allows us to relate the Spencer cohomology with standard constructions in homological algebra and, in particular, to express it as a ..."
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Abstract. We propose a new point of view of the Spencer cohomology appearing in the formal theory of differential equations based on a dual approach via comodules. It allows us to relate the Spencer cohomology with standard constructions in homological algebra and, in particular, to express it as a Cotor. We discuss concrete methods for its construction based on homological perturbation theory. Appears in Georgian Math. J., vol. 9, No. 4, 2002, 723772. 1.
Integrated Development of Algebra in Type Theory
, 1998
"... We present the project of developing computational algebra inside type theory in an integrated way. As a first step towards this, we present direct constructive proofs of Dickson's lemma and Hilbert's basis theorem, and use this to prove the constructive existence of Grobner bases. This can be se ..."
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We present the project of developing computational algebra inside type theory in an integrated way. As a first step towards this, we present direct constructive proofs of Dickson's lemma and Hilbert's basis theorem, and use this to prove the constructive existence of Grobner bases. This can be seen as an integrated development of the Buchberger algorithm, and so far we have a concise formalisation of Dickson's lemma in Half, a type checker for a variant of MartinLof's type theory. We then present work in progress on understanding commutative algebra constructively in type theory using formal topology. Currently we are interested in interpreting existence proofs of prime and maximal ideals, and valuation rings. We give two casestudies: a proof that certain a are nilpotent which uses prime ideals, and a proof of Dedekind's Prague theorem which uses valuation rings. 1 Introduction For the development and formal verification of algorithms, there are essentially two methods [...
PROCEEDINGS OF THE 1992 WORKSHOP ON TYPES FOR PROOFS AND PROGRAMS Bastad
"... The aim of this note is first to set up some general theory for discussing different aspects of the notion of a logic and then to draw attention to the schematic aspects of logic and suggest a way of capturing this aspect without making any commitment to the kind of syntax a logic should have. I ..."
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The aim of this note is first to set up some general theory for discussing different aspects of the notion of a logic and then to draw attention to the schematic aspects of logic and suggest a way of capturing this aspect without making any commitment to the kind of syntax a logic should have. Introduction Nowadays we are well aware that there are many different logics. There are computer systems which are meant to be used to implement many logics. But there is no generally accepted account of what a logic is. Perhaps this is as it should be. We need imprecision in our vocabulary to mirror the flexible imprecision of our thinking. There are a number of related phrases that seem to have a similar imprecision; e.g. formal system, language, axiom system, theory, deductive system, logical system, etc... These are sometimes given technical meanings, often without adequate consideration of the informal notions. When a logic has been implemented in a computer system the logic has been r...