  Mathematical representations of graphs often resemble adjacency matrices or
lists, representations that facilitate whiteboard reasoning and algorithm
design. In the realm of proof assistants, inductive representations effectively
define semantics for formal reasoning. This highlights a gap where algorithm
design and proof assistants require a fundamentally different structure of
graphs, particularly for process theories which represent programs using
graphs. To address this gap, we present VyZX, a verified library for reasoning
about inductively defined graphical languages. These inductive constructs arise
naturally from category theory definitions. A key goal for VyZX is to Verify
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
VyZX are used to prove the correctness of the ZX-calculus rewrite rules and
apply them in practice using standard proof assistant techniques. VyZX
integrates easily with the proof engineer's workflow through visualization and
automation.
