Although computational complexity is a fundamental aspect of program behavior, it is often at odds with common type theoretic principles such as function extensionality, which identifies all functions with the same $textit{input-output}$ behavior. We present a computational type theory called $mathbf{CATT}$ that has a primitive notion of cost (the number of evaluation steps). We introduce a new dependent function type funtime whose semantics can be viewed as a cost-aware version of function extensionality. We prove a collection of lemmas for $mathbf{CATT}$, including a novel introduction rule for the new funtime type. $mathbf{CATT}$ can be simultaneously viewed as a framework for analyzing computational complexity of programs and as the beginnings of a semantic foundation for characterizing feasible mathematical proofs.
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