Reseña

Se ha publicado un artículo de razonamiento formalizado en Coq sobre geometría titulado Parallel postulates and decidability of intersection of lines: a mechanized study within Tarski’s system of geometry.

Sus autores son Pierre Boutry, Julien Narboux y Pascal Schreck (del Équipe Informatique Géométrique et Graphique en la Universidad de Estrasburgo, Francia)

Su resumen es

In this paper we focus on the formalization of the proof of equivalence between different versions of Euclid’s 5 th postulate. This postulate is of historical importance because for centuries many mathematicians believed that this statement was rather a theorem which could be derived from the first four of Euclid’s postulates and history is rich of incorrect proofs of Euclid’s 5 th postulate. These proofs are incorrect because they assume more or less implicitly a statement which is equivalent to Euclid’s 5 th postulate and whose validity is taken for granted. Even though these proofs are incorrect the attempt was not pointless because the flawed proof can be turned into a proof that the unjustified statement implies the parallel postulate. In this paper we provide formal proofs verified using the Coq proof assistant that 10 different statements are equivalent to Euclid’s 5 th postulate. We work in the context of Tarski’s neutral geometry without continuity nor Archimedes’ axiom. The formalization provide a clarification of the hypotheses used for the proofs. Following Beeson, we study the impact of the choice of a particular version of the parallel postulate on the decidability issues.

El código de las correspondientes teorías en Coq se encuentra aquí.

Se ha publicado un artículo de razonamiento formalizado en Coq sobre topología algebraica titulado A formal verification of the theory of parity complexes.

Su autor es Mitchell Buckley (de la Macquarie University en Australia).

Su resumen es

We formalise, in Coq, the opening sections of Parity Complexes up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited by simplexes, cubes and globes, that enable the construction of free ω-categories on such objects. The excision of extremals is a recursive algorithm that presents every cell in such a category as a composite of atomic cells, this is the sense in which the ω-category is free. Due to the complicated multi-dimensional nature of this work, the detail of definitions and proofs can be hard to follow and verify. Indeed, some corrections [Street1994] were required some years following the original publication. Our formalisation verifies that all cases of each result operate as stated. In particular, we indicate which portions of the theory can be proved directly from definitions, and which require more subtle and complex arguments. By identifying results that require the most complicated proofs, we are able to investigate where this theory might benefit from further study and which results need to be considered most carefully in future work.

El código de las correspondientes teorías en Coq se encuentra aquí.