We present a development of Universal Algebra inside Type Theory, formalized using the proof assistant Coq. We define the notion of a signature and of an algebra over a signature. We use setoids, i.e. ...

Abstract. While implementing a proof for the Basic Perturbation Lemma (a central result in Homological Algebra) in the theorem prover Isabelle one faces problems such as the implementation of algebraic structures, partial functions in a logic of total functions, or the level of abstraction in formal proofs. Different approaches aiming at solving these problems will be evaluated and classified according to features such as the degree of mechanization obtained or the direct correspondence to the mathematical proofs. From this study, an environment for further developments in Homological Algebra will be proposed. 1

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 Martin-Lof'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 case-studies: 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 [...

The main purpose of this article is to present a panorama of tools for formal proof analysis and management. The proposed tools are based upon several notions of dependency. Our principal objective is to facilitate the development and the maintenance of theories in proofs assistants in order to improve the productivity of such systems' users.

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