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sav08:herbrand_universe_for_equality [2008/04/02 16:08] vkuncak |
sav08:herbrand_universe_for_equality [2008/04/02 20:45] vkuncak |
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===== Constructing Model for Formulas with Equality ===== | ===== 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. | + | 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]]). |
For each element $t \in GT$, define | For each element $t \in GT$, define | ||
\[ | \[ | ||
- | [t] = \{ s \mid (s,t) \in e_F(I_H)(eq) \} | + | [t] = \{ s \mid (s,t) \in I_H(eq) \} |
\] | \] | ||
Let | Let | ||
Line 25: | Line 25: | ||
[GT] = \{ [t] \mid t \in GT \} | [GT] = \{ [t] \mid t \in GT \} | ||
\] | \] | ||
- | The constructed model will be $([GT],I_Q)$ where $I_Q$ is such that | + | The constructed model will be $([GT],I_Q)$ where |
\[ | \[ | ||
- | I_Q(f)([t_1],\ldots,[t_n]) = [f(t_1,\ldots,t_n)] | + | I_Q(f) = \{ ([t_1],\ldots,[t_n], [f(t_1,\ldots,t_n)]) \mid t_1,\ldots,t_n \in GT \} |
\] | \] | ||
\[ | \[ | ||
I_Q(R) = \{ ([t_1],\ldots,[t_n]) \mid (t_1,\ldots,t_n) \in I_H(R) \} | I_Q(R) = \{ ([t_1],\ldots,[t_n]) \mid (t_1,\ldots,t_n) \in I_H(R) \} | ||
\] | \] | ||
- | The interpretation of eq in $([GT],I_Q)$ becomes equality. | ||
- | Inductive lemma. | + | 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. | ||
+ | ++ | ||
===== Herbrand-Like Theorem for Equality ===== | ===== Herbrand-Like Theorem for Equality ===== |