Abstract
Some parts of dependently typed programs constitute evidence of their type-correctness and, once checked, are unnecessary for execution. These parts can easily become asymptotically larger than the remaining runtime-useful computation, which can cause normally linear-time programs run in exponential time, or worse. We should not make programs run slower by just describing them more precisely. Current dependently typed systems do not erase such computation satisfactorily. By modelling erasure indirectly through type universes or irrelevance, they impose the limitations of these means to erasure. Some useless computation then cannot be erased and idiomatic programs remain asymptotically sub-optimal. In this paper, we explain why we need erasure, that it is different from other concepts like irrelevance, and propose a dependently typed calculus with pattern matching with erasure annotations to model it. We show that erasure in well-typed programs is sound in that it commutes with reduction. Assuming the Church-Rosser property, erasure furthermore preserves convertibility in general. We also give an erasure inference algorithm for erasure-unannotated or partially annotated programs and prove it sound, complete, and optimal with respect to the typing rules of the calculus. Finally, we show that this erasure method is effective in that it can not only recover the expected asymptotic complexity in compiled programs at run time, but it can also shorten compilation times.
Highlights
Consider the following fragment of an Idris program that computes the successor of a binary number
It includes a definition of binary numbers, indexed by their value as natural numbers, and the type signature of add1, which guarantees that the result of add1 must be the successor of the given binary number
We demonstrate the effectiveness of our erasure method by showing the output of erasure on an example, and by showing a few benchmark results
Summary
Consider the following fragment of an Idris program that computes the successor of a binary number. It includes a definition of binary numbers, indexed by their value as natural numbers, and the type signature of add, which guarantees that the result of add must be the successor of the given binary number. What’s worse, in addition to storing these numbers, we need to perform arithmetic on the unary indices, too. This is illustrated using red colour in the (somewhat contrived) function below, which doubles the given binary number and adds one to the result
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