Higher order unification is equational unification for fij-conversion. But it is not first order equational unification, as substitution has to avoid capture. Thus the methods for equational unification (such as narrowing) built upon grafting (i.e. substitution without renaming), cannot be used for higher order unification, which needs specific algorithms. Our goal in this paper is to reduce higher order unification to first order equational unification in a suitable theory. This is achieved by replacing substitution by grafting, but this replacement is not straightforward as it raises two major problems. First, some unification problems have solutions with grafting but no solution with substitution. Then equational unification algorithms rest upon the fact that grafting and reduction commute. But grafting and fij-reduction do not commute in-calculus and reducing an equation may change the set of its solutions. This difficulty comes from the interaction between the substitutions initiated by fij-reduction and the ones initiated by the unification process. Two kinds of variables are involved: those of fij-conversion and those of unification. So, we need to set up a calculus which distinguishes these two kinds of variables and such that reduction and grafting commute. For this purpose, the application of a substitution of a reduction variable to a unification one must be delayed until this variable is instantiated. Such a separation and delay are provided by
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