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.
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