If the terms are composed of different function symbols or the same functor but on different number of terms, the constraint is unsatisfiable.
32.
Completeness can also be proved : if a set of formulae is unsatisfiable, there exists a closed tableau built from it by these rules.
33.
Tableaux with unification can be proved complete : if a set of formulae is unsatisfiable, it has a tableau-with-unification proof.
34.
Even if constraint propagation does not produce an empty domain or an unsatisfiable constraint, it may nevertheless reduce the domains or strengthen the constraints.
35.
The constraint store is unsatisfiable if a variable is bound to take both a value of the specific domain and a functor applied to terms.
36.
In the other way around, a non-uniform problem tells which sets of literals represent satisfiable instances and which ones represent unsatisfiable instances.
37.
This is in general a correct but incomplete unsatisfiability algorithm : a problem may be unsatisfiable even if no empty domain or constraint relation is produced.
38.
Once relations are named, a non-uniform problem expresses a set of sets of literals : those associated to satisfiable ( or unsatisfiable ) instances.
39.
P ( x ) \ } can only be proved unsatisfiable if both P ( a ) and P ( b ) are generated from \ forall x.
40.
As a corollary ( i . e ., its contrapositive ), the compactness theorem says that every unsatisfiable first-order theory has a finite unsatisfiable subset.
