---
title: "Encompassment ordering"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Encompassment_ordering"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:33:08.154025+00:00"
instance: "kb-cron"
---

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

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, nor total
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 with (<) is well-founded, where (<) denotes the irreflexive kernel of (≤). In particular, (<) itself is well-founded.


== Notes ==


== References ==