Schematic Unification

We present a generalization of first-order unification to a term algebra where variable indexing is part of the object language. We exploit variable indexing by associating some sequences of variables ($X_0,X_1,X_2,\dots$) with a mapping $\sigma$ whose domain is the variable sequence and whose range consist of terms that may contain variables from the sequence. From a given term $t$, an infinite sequence of terms may be produced by iterative application of $\sigma$. Given a unification problem $U$ and mapping $\sigma$, the $\mathit{\text{schematic}}\mathit{\text{unification}}\mathit{\text{problem}}$ asks whether all unification problems $U$, $\sigma (U)$, $\sigma (\sigma (U))$, $\dots$ are unifiable. We provide a terminating and sound algorithm. Our algorithm is $\mathit{\text{complete}}$ if we further restrict ourselves to so-called $\mathrm{\infty }$-stable problems. We conjecture that this additional requirement is unnecessary for completeness. Schematic unification is related to methods of inductive proof transformation by resolution and inductive reasoning.