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We present an approach to develop folds for nested data types using dependenttypes. We call such folds $\textit{dependently typed folds}$, they have thefollowing properties. (1) Dependently typed folds are defined by well-foundedrecursion and they can be defined in a total dependently typed language. (2)Dependently typed folds do not depend on maps, map functions and manyterminating functions can be defined using dependently typed folds. (3) Theinduction principles for nested data types follow from the definitions ofdependently typed folds and the programs defined by dependently typed folds canbe formally verified. (4) Dependently typed folds exist for any nested datatypes and they can be specialized to the traditional $\textit{higher-orderfolds}$. Using various of examples, we show how to program and reason aboutdependently typed folds. We also show how to obtain dependently typed folds ingeneral and how to specialize them to the corresponding higher-order folds.

Dependently Typed Folds for Nested Data Types