Abstract
This paper develops a general technique to analyze the head reduction of a term in a context. This technique is used to give a direct proof of the theorem of Hyland and Wadsworth : two λ-terms that have the same Böhm trees, up to (possibly infinite) η-equivalence, are operationally equivalent. It is also used to prove a conjecture of R. Kerth : Every unsolvable λ-term has a decoration. This syntactical result is motivated by (and gives the solution to) a semantical problem.