A graded dependent type system with a usage-aware semantics

Graded Type Theory provides a mechanism to track and reason about resource usage in type systems. In this paper, we develop GraD, novel version of such graded dependent system that includes functions, tensor products, additive sums, unit type. Since standard operational semantics is resource-agnostic, heap-based prove soundness theorem shows correct accounting usage. Several useful properties, including the theorem, non-interference irrelevant resources computation single pointer property for linear resources, can be derived from theorem. We hope our work will provide base integrating linearity, irrelevance types practical programming languages like Haskell.