Interaction Trees: Representing Recursive and Impure Programs in Coq (Work In Progress)
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We present "interaction trees" (ITrees), a general-purpose data structure in Coq for formalizing the behaviors of recursive programs that interact with their environment. ITrees are built of uninterpreted events and their continuations—a coinductive variant of a "free monad". We study the compositional properties of interpreters built from "event handlers" and show how to use them to implement a general mutual recursion operator. The resulting theory enables equational reasoning about ITrees up to weak bisimulation. Using this theory, we prove the termination-sensitive correctness of a compiler from a simple imperative source language to an assembly-like target whose meanings are given as an ITree-based denotational semantics. Crucially, the correctness proof follows entirely by structural induction and the equational theory of combinators for control-flow graphs, which are built on top of the mutual recursion operator. ITrees are also executable, e.g. through extraction, making them suitable for debugging, testing, and implementing executable artifacts.
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