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Formal transformations somehow resembling the usual derivative aresurprisingly common in computer science, with two notable examples beingderivatives of regular expressions and derivatives of types. A newcomer to thislist is the incremental $\lambda$-calculus, or ILC, a "theory of changes" thatdeploys a formal apparatus allowing the automatic generation of efficientupdate functions which perform incremental computation. The ILC is not onlydefined, but given a formal machine-understandable definition---accompanied bymechanically verifiable proofs of various properties, including in particularcorrectness of various sorts. Here, we show how the ILC can be mutated intopropagating tangents, thus serving as a model of Forward Accumulation ModeAutomatic Differentiation. This mutation is done in several steps. These stepscan also be applied to the proofs, resulting in machine-checked proofs of thecorrectness of this model of forward AD.

Evolving the Incremental {\lambda} Calculus into a Model of Forward Automatic Differentiation (AD)