IsaCoSy is a theory formation system for inductive theories. It synthesises conjectures and uses the ones that can be proved to produce a background theory for a new formalisation within a proof assistant. We present a formal account of the algorithms implemented in the system, and prove their correctness. In particular, we show that IsaCoSy only produces irreducible terms, using constraints generated from the left-hand sides of a set of rewrite rules.