Higher-order unification via explicit substitutions

@article{Dowek1995HigherorderUV,
  title={Higher-order unification via explicit substitutions},
  author={Gilles Dowek and Th{\'e}r{\`e}se Hardin and Claude Kirchner},
  journal={Proceedings of Tenth Annual IEEE Symposium on Logic in Computer Science},
  year={1995},
  pages={366-374}
}
Higher-order unification is equational unification for /spl beta//spl eta/-conversion, but it is not first-order equational unification, as substitution has to avoid capture. In this paper higher-order unification is reduced to first-order equational unification in a suitable theory: the /spl lambda//spl sigma/-calculus of explicit substitutions. 
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References

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A rule-based unification procedure in this combined theory is described and may be viewed as an extension of the one initially designed by G. Dowek, T. Hardin and C. Kirchner for performing unification of simply typed λ-terms in a first-order setting via the λσ-calculus of explicit substitutions.

Higher-order unification via explicit substitutions Extended Abstract

TLDR
It is shown that higher-order uni- fication can be reduced to first-order equational unifi- cation in a suitable theory: the A-calculus of explicit substitutions, the kernel of deduction processes used in theorem provers and programming languages.

Implementation of Higher-Order Unification Based on Calculus of Explicit Substitution

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This paper presents several improvements of an algorithm for a higher-order unification based on the calculus of explicit substitutions that tries to postpone normalisation of λσ-terms as long as possible, i.e. until some information is necessary for the next step of the unification algorithm.

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In this paper, several improvements of an algorithm for a higher-order uniication based on the calculus of explicit substitutions are presented, that tries to postpone normalisation of-terms as long as possible, i.e. until some information of these-terms is necessary for the next step of the uniications algorithm.

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