Holes with Binding Power

Incomplete logical proofs are the logical counterpart of the incomplete λ-terms that one usually works with in an interactive theorem prover based on type theory. In this paper we extend the formalization of such incomplete proofs given in [5] by introducing unknowns that are allowed to provide temporary bindings for variables that are supposed to be bound, but whose binders are not constructed yet a situation that typically occurs when one does forward reasoning. We do this by introducing hereditarily parameterized meta-variables and show that by separating the object-level from the meta-level abstractions one can get the abstraction needed to implement the incomplete objects without having to extend the function spaces of the object-level system. We define a typing system that extends λHOL and re-establish the formulas-as-types embedding of the logic with binding holes into this system.