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sav08:herbrand_universe_for_equality [2008/04/02 01:03] vkuncak |
sav08:herbrand_universe_for_equality [2008/04/02 21:43] vkuncak |
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call the resulting formula $F'$. | call the resulting formula $F'$. | ||
- | Consider Herbrand model $I_H$ for $\{F'\} \cup AxEq$. The relation $e_F(I_H)(eq)$ splits $GT$ into two sets: the set of terms eq with $a$, and the set of terms eq with $b$. | + | Consider Herbrand model $I_H = (GT,\alpha_H)$ for the set $\{F'\} \cup AxEq$. The relation $\alpha_H(eq)$ splits $GT$ into two sets: the set of terms eq with $a$, and the set of terms eq with $b$. The idea is to consider these two partitions as domain of new interpretation, denoted $([GT], \alpha_Q)$. |
===== 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 |
- | For each element $t \in GT$, define | ||
- | \[ | ||
- | [t] = \{ s \mid (s,t) \in e_F(I_H)(eq) \} | ||
- | \] | ||
- | Let | ||
- | \[ | ||
- | [GT] = \{ [t] \mid t \in GT \} | ||
- | \] | ||
- | The constructed model will have $[GT]$ as the domain. We define its interpretation $I_Q$ so that | ||
- | \[ | ||
- | I_Q(f)([t_1],\ldots,[t_n]) = [f(t_1,\ldots,t_n)] | ||
- | \] | ||
- | \[ | ||
- | I_Q(R) = \{ ([t_1],\ldots,[t_n]) \mid (t_1,\ldots,t_n) \in I_H(R) \} | ||
- | \] | ||
===== Herbrand-Like Theorem for Equality ===== | ===== Herbrand-Like Theorem for Equality ===== | ||
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**Theorem:** For every set of formulas with equality $S$ the following are equivalent | **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 | + | * $S' \cup AxEq$ 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 of ground terms under some congruence | + | * $S$ has a model whose domain is the quotient $[GT]$ of ground terms under some congruence |