Encompassment ordering

Triangle diagram of two terms s ≤ t related by the encompassment preorder.

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

References