Graphical languages are a convenient shorthand to represent computation, with rewrite rules relating one graph to another. In contrast, proof assistants rely heavily on inductive datatypes, particularly when giving semantics to embedded languages. This creates obstacles to formally reasoning about graphical languages, since imposing an inductive structure obfuscates the diagrammatic nature of graphical languages, along with their corresponding equational theories. To address this gap, we present Vy ZX, a verified library for reasoning about inductively defined graphical languages. These inductive constructs arise naturally from category-theoretic definitions. We developed Vy ZX to V erify the ZX -calculus, a graphical language for reasoning about quantum computation. The ZX-calculus comes with a collection of diagrammatic rewrite rules that preserve the graph’s semantic interpretation. We show how inductive graphs in Vy ZX are used to prove the soundness of the ZX-calculus rewrite rules and apply them in practice using standard proof assistant techniques. We also provide an IDE-integrated visualizer for proof engineers to directly reason about diagrams in graphical form.

@article{paperbot3952,
  title = {<i>Vy</i>
                    ZX: Formal Verification of a Graphical Quantum Language},
  author = {Adrian Lehmann and Ben Caldwell and Bhakti Shah and William Spencer and Robert Rand},
  journal = {ACM Transactions on Programming Languages and Systems},
  volume = {48},
  number = {3},
  year = {2026},
  doi = {10.1145/3807780}
}