CHAD for expressive total languages

We show how to apply forward and reverse mode Combinatory Homomorphic Automatic Differentiation (CHAD) (Vákár (2021). ESOP , 607–634; Vákár and Smeding (2022). ACM Transactions on Programming Languages and Systems 44 (3) 20:1–20:49.) to total functional programming languages with expressive type systems featuring the combination of • tuple types; • sum types; • inductive types; • coinductive types; • function types. We achieve this by analyzing the categorical semantics of such types in Σ -types (Grothendieck constructions) of suitable categories. Using a novel categorical logical relations technique for such expressive type systems, we give a correctness proof of CHAD in this setting by showing that it computes the usual mathematical derivative of the function that the original program implements. The result is a principled, purely functional and provably correct method for performing forward- and reverse-mode automatic differentiation (AD) on total functional programming languages with expressive type systems.