Abstract. In this paper, a proof assistant, called SAD, is presented. SAD deals with mathematical texts that are formalized in the ForTheL language (brief description of which is also given) and checks their correctness. We give a short description of SAD and a series of examples that show what can be done with it. Note that abstract notion of correctness on which the implementation is based, can be formalized with the help of a calculus (not presented here).

Abstract.Formalizing mathematical argument is a fascinating activity in itself and (we hope!) also bears important practical applications. While traditional proof theory investigates deducibility of an individual statement from a collection of premises, a mathematical proof, with its structure and continuity, can hardly be presented as a single sequent or a set of logical formulas. What is called "mathematical text", as used in mathematical practice through the ages, seems to be more appropriate. However, no commonly adopted formal notion of mathematical text has emerged so far.In this paper, we propose a formalism which aims to reflect natural (human) style and structure of mathematical argument, yet to be appropriate for automated processing: principally, verification of its correctness (we consciously use the word rather than "soundness" or "validity").We consider mathematical texts that are formalized in the ForTheL language (brief description of which is also given) and we formulate a point of view on what a correct mathematical text might be. Logical notion of correctness is formalized with the help of a calculus. Practically, these ideas, methods and algorithms are implemented in a proof assistant called SAD. We give a short description of SAD and a series of examples showing what can be done with it.

No abstract

The System for Automated Deduction (SAD) is developed in the framework of the Evidence Algorithm research project and is intended for automated processing of mathematical texts. The SAD system works on three levels of reasoning: (a) the level of text presentation where proofs are written in a formal natural-like language for subsequent verification; (b) the level of foreground reasoning where a particular theorem proving problem is simplified and decomposed; (c) the level of background deduction where exhaustive combinatorial inference search in classical first-order logic is applied to prove end subgoals.We present an overview of SAD describing the ideas behind the project, the system's design, and the process of problem formalization in the fashion of SAD. We show that the choice of classical first-order logic as the background logic of SAD is not too restrictive. For example, we can handle binders like Σ or lim without resort to second order or to a full-powered set theory. We illustrate our approach with a series of examples, in particular, with the classical problem √ 2 / ∈ Q.

New sequent forms * of the famous Herbrand theorem are proved for first-order classical logic without equality. These forms use the original notion of an admissible substitution and a certain modification of the Herbrand universe, which is constructed from constants, special variables, and functional symbols occurring only in the signature of an initial theory. Other well-known forms of the Herbrand theorem are obtained as special cases of the sequent ones. Besides, the sequent forms give an approach to the construction and theoretical investigation of computer-oriented calculi for efficient logical inference search in the signature of an initial theory. In a comparably simple way, they provide us with some technique for proving the completeness and soundness of the calculi.