Include η-reduction in interaction improvement

Determine whether η-reduction is included in the interaction improvement preorder Eint-imp on ordinary untyped lambda-terms (defined via black lifting into the checkers calculus), i.e., prove that for all terms t and u with t →η u, it holds that t Eint-imp u. This would establish that the inclusion Eint ⊆ Eint-imp is strict.

Background

The paper introduces interaction improvement (Eint-imp) as a preorder refining contextual reasoning by counting interaction steps while ignoring silent ones, in the checkers calculus and then transferring it to ordinary λ-terms via black lifting. While the interaction preorder Eint is shown to be an inequational λ-theory, its relationship with η-reduction remains unresolved.

The authors argue that if η-reduction were included in Eint-imp, it would strictly extend Eint, reflecting that η-reduction can decrease the number of interactions in head evaluations (as illustrated by their examples). However, standard rewriting techniques and their semantic tools do not presently handle η within the checkers framework.