# Encompassment ordering

In theoretical computer science, in particular in automated theorem proving and term rewriting,
the **containment**,[1] or **encompassment**, preorder (≤) on the set of terms, is defined by[2]

*s*≤*t*if a subterm of*t*is a substitution instance of*s*.

It is used e.g. in the Knuth–Bendix completion algorithm.

## Properties

- Encompassment is a preorder, i.e. reflexive and transitive, but not anti-symmetric,[note 1] nor total[note 2]
- The corresponding equivalence relation, defined by
*s*~*t*if*s*≤*t*≤*s*, is equality modulo renaming. *s*≤*t*whenever*s*is a subterm of*t*.*s*≤*t*whenever*t*is a substitution instance of*s*.- The union of any well-founded rewrite order
*R*[note 3] with (<) is well-founded, where (<) denotes the irreflexive kernel of (≤).[3] In particular, (<) itself is well-founded.

## Notes

- since both
*f*(*x*) ≤*f*(*y*) and*f*(*y*) ≤*f*(*x*) for variable symbols*x*,*y*and a function symbol*f* - since neither
*a*≤*b*nor*b*≤*a*for distinct constant symbols*a*,*b* - i.e. irreflexive, transitive, and well-founded binary relation
*R*such that*sRt*implies*u*[*s*σ]_{p}R*u*[*t*σ]_{p}for all terms*s*,*t*,*u*, each path*p*of*u*, and each substitution*σ*

## References

- Gerard Huet (1981). "A Complete Proof of Correctness of the Knuth–Bendix Completion Algorithm".
*J. Comput. Syst. Sci*.**23**(1): 11–21. doi:10.1016/0022-0000(81)90002-7. - N. Dershowitz, J.-P. Jouannaud (1990). Jan van Leeuwen (ed.).
*Rewrite Systems*. Handbook of Theoretical Computer Science.**B**. Elsevier. pp. 243–320. Here:sect.2.1, p. 250 - Dershowitz, Jouannaud (1990), sect.5.4, p. 278; somewhat sloppy,
*R*is required to be a "terminating rewrite relation" there.

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