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--- sav08:herbrand_universe_for_equality [2008/04/02 21:05]
vkuncak
+++ sav08:herbrand_universe_for_equality [2008/04/02 22:59]
vkuncak
@@ Line 15: / Line 15: @@
 ===== Constructing Model for Formulas with Equality =====
-Let $S$ be a set of formulas in first-order logic with equality and $S'$ result of replacing '=' with 'eq' in $S$.  Suppose that $S \cup AxEq$ is satisfiable.  Let $(GT,I_H)$ be Herbrand model for $S \cup AxEq$.  We construct a new model using //quotient// construction, described as follows (recall notation in [[First-Order Logic Semantics]]).
+Let $S$ be a set of formulas in first-order logic with equality and $S'$ result of replacing '=' with 'eq' in $S$.  Suppose $S' \cup AxEq$ is satisfiable.
-For each element $t \in GT$, define
-\[
-    [t] = \{ s \mid (s,t) \in I_H(eq) \}
-\]
-Let
-\[
-    [GT] = \{ [t] \mid t \in GT \}
-\]
-The constructed model will be $([GT],I_Q)$ where
-\[
-    I_Q(f) = \{ ([t_1],\ldots,[t_n], [f(t_1,\ldots,t_n)]) \mid t_1,\ldots,t_n \in GT \}
-\]
-\[
-    I_Q(R) = \{ ([t_1],\ldots,[t_n]) \mid (t_1,\ldots,t_n) \in I_H(R) \}
-\]
-In particular, when $R$ is $eq$ we have
-$I_Q(eq) = $ ++| $\{ ([t_1],[t_2]) \mid (t_1,t_2) \in I_H(eq) \} = \{ (a,a) \mid a \in [GT] \}$
-that is, the interpretation of eq in $([GT],I_Q)$ is equality.
-++
+Let $I_H = (GT,\alpha_H)$ be Herbrand model for $S \cup AxEq$.  We construct a new model as [[Interpretation Quotient Under Congruence]] of $(GT,\alpha)$ under congruence 'eq'.  Denote quoient structure by $I_Q = ([GT],\alpha_Q)$.  By theorem on quotient structures, $S$ is true in $I_Q$.  Therefore, $S$ is satisfiable.
 ===== Herbrand-Like Theorem for Equality =====
 **Theorem:** For every set of formulas with equality $S$ the following are equivalent
-  * $S$ has a model
+  - $S$ has a model;
-  * $S' \cup AxEq$ has a model (where $AxEq$ are [[Axioms for Equality]] and $S'$ is result of replacing '=' with 'eq' in $S$)
+  - $S' \cup AxEq$ has a model (where $AxEq$ are [[Axioms for Equality]] and $S'$ is result of replacing '=' with 'eq' in $S$);
-  * $S$ has a model whose domain is the quotient $[GT]$ of ground terms under some congruence
+  - $S$ has a model whose domain is the quotient $[GT]$ of the set of ground terms under some congruence.