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sav08:herbrand_universe_for_equality [2008/04/02 20:40] vkuncak |
sav08:herbrand_universe_for_equality [2008/04/02 21:05] vkuncak |
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call the resulting formula $F'$. | call the resulting formula $F'$. | ||
- | Consider Herbrand model $(GT,I_H)$ for the set $\{F'\} \cup AxEq$. The relation $I_H(eq)$ 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 (recall notation in [[First-Order Logic Semantics]]). |
For each element $t \in GT$, define | For each element $t \in GT$, define | ||
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* $S' \cup AxEq$ 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 ground terms under some congruence | ||
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