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

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

References

1. Gerard Huet (1981). "A Complete Proof of Correctness of the Knuth–Bendix Completion Algorithm". J. Comput. System Sci. 23 (1): 11–21. doi:10.1016/0022-0000(81)90002-7.
2. 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
3. Dershowitz, Jouannaud (1990), sect.5.4, p. 278; somewhat sloppy, R is required to be a "terminating rewrite relation" there.