A Vernacular for Coherent Logic

We propose a simple, yet expressive proof representation from which proofs for different proof assistants can easily be generated. The representation uses only a few inference rules and is based on a frag- ment of first-order logic called coherent logic. Coherent logic has been recognized by a number of researchers as a suitable logic for many ev- eryday mathematical developments. The proposed proof representation is accompanied by a corresponding XML format and by a suite of XSL transformations for generating formal proofs for Isabelle/Isar and Coq, as well as proofs expressed in a natural language form (formatted in LATEX or in HTML). Also, our automated theorem prover for coherent logic exports proofs in the proposed XML format. All tools are publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014

Proof Simplification in the Framework of Coherent Logic

The problem of proof simplification draws a lot of attention to itself across various contexts. In this paper, we present one approach for simplifying proofs constructed in the framework of coherent logic. This approach is motivated by the need for filtering-out "clean'' and short proofs from proof-traces, which typically contain many irrelevant steps, and which are generated by automated theorem provers - in this case, theorem provers based on coherent logic. Such "clean'' proofs can then be used for producing readable proofs in natural-language form. The proof simplification procedure consists of three transformation steps. The first one is based on the elimination of inference steps which are irrelevant for the present proof, also allowing some irrelevant branchings to be eliminated, the second one consists of lifting-up steps through the branching steps, followed by elimination of repeated steps, while the third one serves to convert proof fragments into the reductio ad absurdum form, if possible. In contrast to general simplification procedures, our proof simplification procedure is specific for a fragment of first order logic and therefore simple and easy to implement, and allows simple generation of object level proofs. We proceed to prove that this procedure is correct and terminating, and also that it never increases the size of a proof. Finally, we implement the proof simplification procedure, and provide several example proofs

Automated generation of machine verifiable and readable proofs: A case study of Tarski’s geometry

The power of state-of-the-art automated and interactive theorem provers has reached the level at which a significant portion of non-trivial mathematical contents can be formalized almost fully automatically. In this paper we present our framework for the formalization of mathematical knowledge that can produce machine verifiable proofs (for different proof assistants) but also human-readable (nearly textbook-like) proofs. As a case study, we focus on one of the twentieth century classics – a book on Tarski’s geometry. We tried to automatically generate such proofs for the theorems from this book using resolution theorem provers and a coherent logic theorem prover. In the first experiment, we used only theorems from the book, in the second we used additional lemmas from the existing Coq formalization of the book, and in the third we used specific dependency lists from the Coq formalization for each theorem. The results show that 37 % of the theorems from the book can be automatically proven (with readable and machine verifiable proofs generated) without any guidance, and with additional lemmas this percentage rises to 42 %. These results give hope that the described framework and other forms of automation can significantly aid mathematicians in developing formal and informal mathematical knowledge

Formalization and automation of Euclidean geometry

Напредак геометрије кроз векове се може разматрати кроз развој различитих аксиоматских система који је описују. Употреба аксиоматских система започиње са Хилбертом и Тарским али се ту не завршава. Чак и данас се развијају нови аксиоматски ситеми за рад са еуклидском геометријом...The advance of geometry over the centuries can be observed through the development of dierent axiomatic systems that describe it. The use of axiomatic systems begins with Euclid, continues with Hilbert and Tarski, but it doesn't end there. Even today, new axiomatic systems for Euclidean geometry are developed..