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Herbrand Universe for Equality
Recall example of formula :
\[
P(a) \land (\forall x.\ P(x) \rightarrow \lnot P(f(x))) \land (\forall x. x=a \lor x=b)
\] Replace '=' with 'eq': \[
P(a) \land (\forall x.\ P(x) \rightarrow \lnot P(f(x))) \land (\forall x. eq(x,a) \lor eq(x,b))
\]
call the resulting formula .
Consider Herbrand model for the set
. The relation
splits
into two sets: the set of terms eq with
, and the set of terms eq with
. The idea is to consider these two partitions as domain of new interpretation, denoted
.
Constructing Model for Formulas with Equality
Let be a set of formulas in first-order logic with equality and
result of replacing '=' with 'eq' in
. Suppose
is satisfiable.
Let be Herbrand model for
. We construct a new model as Interpretation Quotient Under Congruence of
under congruence 'eq'. Denote quoient structure by
. By theorem on quotient structures,
is true in
. Therefore,
is satisfiable.
Herbrand-Like Theorem for Equality
Theorem: For every set of formulas with equality the following are equivalent
has a model;
has a model whose domain is the quotient
of the set of ground terms under some congruence.