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Encompassment ordering

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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

s≤t if a subterm of t is a substitution instance of s.

It is a preorder, i.e. reflexive and transitive, but not anti-symmetric,[note 1], nor total[note 2] [2] It is used e.g. in the Knuth-Bendix completion algorithm. σ

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

References

  1. ^ Gerard Huet (1981). "A Complete Proof of Correctness of the Knuth-Bendix Completion Algorithm". J. Comput. System Sci. 23 (1): 11–21.
  2. ^ N. Dershowitz, J.-P. Jouannaud (1990). Jan van Leeuwen (ed.). Rewrite Systems. Handbook of Theoretical Computer Science. Vol. B. Elsevier. pp. 243–320. Here:sect.2.1, p.250
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