We announce a tool for mapping derivations of the E theorem prover to Mizar proofs. Our mapping complements earlier work that generates problems for automated theorem provers from Mizar inference checking problems. We describe the tool, explain the mapping, and show how we solved some of the difficulties that arise in mapping proofs between different logical formalisms, even when they are based on the same notion of logical consequence, as Mizar and E are (namely, first-order classical logic with identity).
We do not intend to enter into a discussion about the proof identity problem. For a discussion, see Do s en @cite_15 . Certainly the intension behind the mapping is to preserve whatever abstract proof expressed by the derivation. That the derivation and the text generated from it are isomorphic will be clarified by the discussion below of the translation algorithm. Mapping such as the one discussed here can help contribute to a concrete investigation of the proof identity problem, which in fact motivates the project reported here. The reader need not share the author's interest in the proof identity problem to understand what follows.
@cite_15
2068169489
Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz's, based on reduction to cut-free form in sequent systems, but he also suggested understanding identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to a renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to low-dimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic. ?1. General proof theory. When a question is not in the main stream of mathematical investigations, and is also of a certain general, conceptual, kind, it runs the danger of being dismissed as "philosophical". Before the advent of recursion theory, many mathematicians would presumably have dismissed the question "What is a computable function?" as a philosophical question (or perhaps even as a psychological, empirical, question). It required something like the enthusiasm of a young discipline on the rise, which logic was in the first half of the twentieth century, for such a question to be embraced as legitimate, and seriously treated by mathematical means-with excellent results. An outsider might suppose that the question "What is a proof?" should be important for a field called proof theory, and then he would be surprised to find that this and related questions, one of which will occupy us here, Received on May 27, 2003; revised July 4, 2003; revised August 26, 2003. 2000 Mathematics Subject Classification. 03F03, 03F07, 03A05, 03-03.