A logically saturated extension of $\bar{\lambda} \mu \tilde{\mu}$

This paper presents a proof language based on the work of Sacerdoti Coen [1,2], Kirchner [3] and Autexier [4] on ¯¿µ˜µ, a calculus introduced by Curien and Herbelin [5,6]. Just as ¯¿µ˜µ preserves several proof structures that are identified by the ¿-calculus, the proof language presented here aims to preserve as much proof structure as reasonable; we call that property being logically saturated. This leads to several advantages when the language is used as a generic exchange language for proofs, as well as for other uses. We equip the calculus with a simple rendering in pseudo-natural language that aims to give people tools to read, understand and exchange terms of the language. We show how this rendering can, at the cost of some more complexity, be made to produce text that is more natural and idiomatic, or in the style of a declarative proof language like Isar or Mizar.