An approximation of reductional equivalence

We deene an equivalence relation on-terms called shuue-equivalence which attempts to capture the notion of reductional equivalence on strongly normalizing terms. The shuue-equivalence classes are shown to divide the classes of-equal terms into smaller ones consisting of terms with similar reduction behaviour. We reene-reduction from a relation on terms to a relation on shuue-equivalence classes, called shuue-reduction, and show that this reenement captures existing generalisations of-reduction. Shuue-reduction moreover, apart from allowing one to make more redexes visible and to contract these newly visible redexes, enables one to have more freedom in choosing the reduction path of a term, which can result in smaller terms along the reduction path if a clever reduction strategy is used. This can beneet both programming languages and theorem provers since this exibility and freedom in chosing reduction paths can be exploited to produce the shortest program evaluation paths and optimal proofs.