Inductive Reasoning for Coinductive Types

Abstract

We present AlgCo (Algebraic Coinductives), a practical framework for inductive reasoning over commonly used coinductive types such as conats, streams, and infinitary trees with finite branching factor. The key idea is to exploit the domain-theoretic notion of algebraic CPO to define continuous operations over coinductive types indirectly via primitive recursion on "dense" collections of their elements. This enables a convenient strategy for reasoning about algebraic coinductives by straightforward proofs by induction. We implement the AlgCo framework in Coq and demonstrate its power by verifying a stream variant of the sieve of Eratosthenes, a regular expression library based on coinductive trie encodings of formal languages, and weakest pre-expectation style semantics for coinductive sampling processes over discrete probability distributions in the random bit model.