Unraveling the iterative CHAD

Combinatory Homomorphic Automatic Differentiation (CHAD) was originally formulated as a semantics-driven source transformation for reverse-mode AD in total programming languages. We extend this framework to partial languages with features such as potentially non-terminating operations, real-valued conditionals, and iteration constructs like while-loops, while preserving CHAD's structure-preserving semantics principle. A key contribution is the introduction of iteration-extensive indexed categories, which allow iteration in the base category to lift to parameterized initial algebras in the indexed category. This enables iteration to be interpreted in the Grothendieck construction of the target language in a principled way. The resulting fibred iterative structure cleanly models iteration in the categorical semantics. Consequently, the extended CHAD transformation remains the unique structure-preserving functor (an iterative Freyd category morphism) from the freely generated iterative Freyd category of the source language to the Grothendieck construction of the target's syntactic semantics, mapping each primitive operation to its derivative. We prove the correctness of this transformation using the universal property of the source language's syntax, showing that the transformed programs compute correct reverse-mode derivatives. Our development also contributes to understanding iteration constructs within dependently typed languages and categories of containers. As our primary motivation and application, we generalize CHAD to languages with data types, partial features, and iteration, providing the first rigorous categorical semantics for reverse-mode CHAD in such settings and formally guaranteeing the correctness of the source-to-source CHAD technique.