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.



  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


  1. 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.
  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.

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