Equational reasoning with string diagrams provides an intuitive means of
proving equations between morphisms in a symmetric monoidal category. This can
be extended to proofs of infinite families of equations using a simple
graphical syntax called !-box notation. While this does greatly increase the
proving power of string diagrams, previous attempts to go beyond equational
reasoning have been largely ad hoc, owing to the lack of a suitable logical
framework for diagrammatic proofs involving !-boxes. In this paper, we extend
equational reasoning with !-boxes to a fully-fledged first order logic called
with conjunction, implication, and universal quantification over !-boxes. This
logic, called !L, is then rich enough to properly formalise an induction
principle for !-boxes. We then build a standard model for !L and give an
example proof of a theorem for non-commutative bialgebras using !L, which is
unobtainable by equational reasoning alone.Comment: 15 pages + appendi
