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.
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
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- ↑ Lua error in package.lua at line 80: module 'strict' not found. Here:sect.2.1, p. 250
- ↑ Dershowitz, Jouannaud (1990), sect.5.4, p. 278; somewhat sloppy, R is required to be a "terminating rewrite relation" there.
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