Summary: Proof terms in term rewriting are a representation means for reduction sequences, and more in general for contraction activity, allowing to distinguish e.g. simultaneous from sequential reduction. Proof terms for finitary, first-order, left-linear term rewriting are described in the literature. In a previous work, we defined an extension of the finitary proof-term formalism, that allows to describe contractions in infinitary first-order term rewriting, and gave a characterisation of permutation equivalence.
In this work, we discuss how projections of possibly infinite rewrite sequences can be modelled using proof terms. Again, the foundation is a characterisation of projections for finitary rewriting. We extend this characterisation to infinitary rewriting and also refine it, by describing precisely the role that structural equivalence plays in the development of the notion of projection. The characterisation we propose yields a definite expression, i.e. a proof term, that describes the projection of an infinitary reduction over another.
To illustrate the working of projections, we show how a common reduct of a (possibly infinite) reduction and a single step that makes part of it can be obtained via their respective projections. We show, by means of several examples, that the proposed definition yields the expected behaviour also in cases beyond those covered by this result. Finally, we discuss how the notion of limit is used in our definition of projection for infinite reduction.