We propose a new approach to text generation from formal proofs that exploits the high-level and interactive features of a tactic-style theorem prover. The design of our system is based on communication conventions identified in a corpus of texts. We show how to use dialogue with the theorem prover to obtain information that is required for communication but is not explicitly used in reasoning.

There are two dierent approaches to formalizing proofs in a computer: the procedural approach (which is the one of the HOL system) and the declarative approach (which is the one of the Mizar system).

We present a complete formalization of the Hahn-Banach theorem in the simply-typed set-theory of Isabelle/HOL, such that both the modeling of the underlying mathematical notions and the full proofs are intelligible to human readers. This is achieved by means of the Isar environment, which provides a framework for high-level reasoning based on natural deduction. The final result is presented as a readable formal proof document, following usual presentations in mathematical textbooks quite closely. Our case study demonstrates that Isabelle/Isar is capable to support this kind of application of formal logic very well, while being open for an even larger scope.

We describe some of the complications involved in expressing the technique of induction when automatically generating textual versions of formal mathematical proofs produced by a theorem proving system, and describe our approach to this problem. The pervasiveness of induction within mathematical proofs makes its effective generation vital to readable proof texts. Our focus is on planning texts involving induction. Our efforts are driven by a corpus of human-produced proof texts, incorporating both regularities within this corpus and the formal structure of induction into coherent text plans.

Is it possible to create formal proofs of interesting mathematical theorems which are mechanically checked in every detail and yet are readable and even faithful to the best expositions of those results in the literature? This paper answers that question positively for theorems about decidable properties of nite automata. The exposition is from Hopcroft and Ullman's classic 1969 textbook Formal Languages and Their Relation to Automata. This paper describes a successful formalization which is faithful to that book. The requirement of being faithful to the book has unexpected consequences, namely that the underlying formal theory must include primitive notions of computability. This requirement makes a constructive formalization especially suitable. It also opens the possibility ofusingthe formal proofs to decide properties of automata. The paper shows how to do this. 1

In the presence of growing collections of formal mathematics, and renewed interest in formal mathematics and automated theorem proving for new domains such as hardware or code verification, it is vital to be able to present formal content accessibly to broad audiences. We propose a novel approach to constructing a content planner for formal mathematics produced by a tactic-style prover which capitalizes on the inherent structure of the formal proofs. Though it had been posited that high-level formal structure is unsuitable as a source of information for text generation, due to its heuristic nature and necessary lack of details, we are able to show that this is not the case. Tactic-style proofs share significant structural commonality with the discourse structure of corresponding texts. These commonalities allow a content planner to be constructed which need only use low-level logical content as a supplementary information source to the generation process. To show that this is the case, we collected two corpora of texts generated to communicate the proof content of a series of formal proofs produced by the Nuprl