This paper solves problem #79 of RTA’s list of open problems [14] — in the positive. If the
rules of a TRS do not overlap w.r.t. substitutions of infinite terms then the TRS has unique
normal forms. We solve the problem by reducing the problem to one of consistency for “similar”
constructor term rewriting systems. For this we introduce a new proof technique. We define a
relation ⇓ that is consistent by construction, and which — if transitive — would coincide with
the rewrite system’s equivalence relation =R.
We then prove the transitivity of ⇓ by coalgebraic reasoning. Any concrete proof for instances
of this relation only refers to terms of some finite coalgebra, and we then construct an equivalence
relation on that coalgebra which coincides with ⇓
