# 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.

## 161 Citations

### Higher-order unification via explicit substitutions Extended Abstract

- Computer ScienceLICS 1995
- 1995

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.

### From Higher-Order to First-Order Rewriting

- Computer ScienceRTA
- 2001

A characterization of the class of higher-order rewriting systems which can be encoded by first- order rewriting modulo an empty theory (that is, Ɛ = θ), which includes of course the λ-calculus.

### From Higher-Order to First-Order Rewriting ( Extended Abstract )

- Computer Science
- 2001

A characterization of the class of higher-order rewriting systems which can be encoded by first- order rewriting modulo an empty theory (that is, E = ∅) is obtained, which includes of course the λ-calculus.

### HOL-lambdasigma: An Intentional First-Order Expression of Higher-Order Logic

- PhilosophyRTA
- 1999

A first-order presentation of higher-order logic based on explicit substitutions, i.e. a proposition can be proved without the extensionality axioms in one theory if and only if it can in the other, is proposed.

### Extending Higher-Order Unification to Support Proof Irrelevance

- Computer ScienceTPHOLs
- 2003

This work describes this extended algorithm, whose presentation is simplified by making use of recent developments in explaining unification metavariables as modal variables, which obviates the need for full explicit substitutions.

### Simplifying the signature in second-order unification

- Computer ScienceApplicable Algebra in Engineering, Communication and Computing
- 2009

This work reduces Second-Order Unification to Second-order Unification with a signature that contains just one binary function symbol and constants, based on partially currying the equations by using the binaryfunction symbol for explicit application @.

### Unification for \lambda -calculi Without Propagation Rules

- Computer ScienceICTAC
- 2016

The unification procedure was developed for the calculi with ES that belong to the paradigm known as “act at a distance”, and explicit substitutions are not propagated to the level of variables, as usual.

### Absolute Explicit Unification

- Computer ScienceRTA
- 2000

The system allows to solve type checking, type inhabitation, higher-order unification, and type inference for PTS using purely first-order machinery and a novel feature of the system is that it combines substitutions and variable declarations.

### X.R.S : Explicit Reduction Systems - A First-Order Calculus for Higher-Order Calculi

- Mathematics, Computer ScienceCADE
- 1998

The σ⇑-calculus is used as the substitution mechanism of general higher-order systems which the authors will name Explicit Reduction Systems and general conditions to define a confluent XRS are given.

### Higher-order matching modulo (super)developements. Applications to second-order matching

- Computer Science, Mathematics
- 2008

This work considers higher-order matching modulo (super)developments over untyped lambda-terms for which it proposes terminating, sound and complete matching algorithms and proposes a restriction to second- order matching that gives exactly all second-order matches.

## 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

- Computer ScienceLICS 1995
- 1995

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.

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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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