Linear Contextual Modal Type Theory

When one develops, implements, and studies type theories based on linear logic, for example, in the context of theorem proving, logic programming, and formal reasoning, one is immediately confronted with questions about their equational theory and how to deal with logic variables. In this paper, we propose linear contextual modal type theory that gives a mathematical account of the nature of logic variables. Our type theory is conservative over intuitionistic contextual modal type theory proposed by Nanevski, Pfenning, and Pientka. As a technical contribution we provide a proof of soundness, and, as a justification for its usefulness, we shed some light on the difficulties working with logic variables in linear logics that contain >.