A Simply Typed λ-Calculus of Forward Automatic Differentiation

We present an extension of the simply typed λ-calculus with pushforward operators. This extension is motivated by the desire to incorporate forward automatic differentiation, which is an important technique in numeric computing, into functional programming. Our calculus is similar to Ehrhard and Regnierʼs differential λ-calculus, but is based on the differential geometric idea of pushforward rather than derivative. We prove that, like the differential λ-calculus, our calculus can be soundly interpreted in differential λ-categories.