Towards a Formal Mathematical Vernacular  
authors  Groote, J.F. 
source  Logic Group Preprint Series, volume: 84 (2008) 
full text  [Full text] 
document type  Preprint 
discipline 

abstract  Contemporary proof verificators often use a command language to construct proofs. These commands
are often called tactics. This new generation of theorem provers is a substantial improvement over earlier
ones such as AUTOMATH. Based on experience with these new provers we feel the need to study these
languages further, especially, because we think that these may be improved in their adequateness to express
proofs closer to the established mathematical vernacular. We also feel that a systematic treatment of these
vernaculars may lead to an improvement towards the automatic inference of trivial proof steps. In any
case a systematic treatment will lead to a better understanding of the command languages.
This exercise is carried out in the setting of Pure Type Systems (PTSs) in which a whole range of
logics can be embedded. We first identify a subclass of PTSs, called the PTSs for logic. For this class we define a formal mathematical vernacular and we prove elementary sound and completeness. Via an
elaborate example we try to assess how easy proofs in mathematics can be written down in our vernacular along the lines of the original proofs. 