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 Here:sect.2.1, p. 250 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,since both f(x) ≤ f(y) and f(y) ≤ f(x) for variable symbols x, y and a function symbol f nor totalsince neither a ≤ b nor b ≤ a for distinct constant symbols a, b 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 Ri.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 with (<) is well-founded, where (<) denotes the irreflexive kernel of (≤).Dershowitz, Jouannaud (1990), sect.5.4, p. 278; somewhat sloppy, R is required to be a "terminating rewrite relation" there. In particular, (<) itself is well-founded.
Notes
References