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--- sav08:combining_fol_models [2009/05/13 10:21]
vkuncak
+++ sav08:combining_fol_models [2009/05/13 10:48]
vkuncak
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 ===== Merging Models of Theories with Disjoint Languages (Signatures) =====
-**Theorem:** Let ${\cal L}$ be a finite language ${\cal L} = {\cal L}_1 \cup {\cal L}_2$ where ${\cal L}_1 \cap {\cal L}_2 = \emptyset$.  Let $C$ be a conjunction in language ${\cal L}$, and for $i\in \{1,2\}$ let $C_i$ be those literals from $C$ whose symbols appear only in ${\cal L}_i$, as well as equality and disequality literals.  Suppose further that if variables $x,y \in FV(C_1)\cap FV(C_2)$ then either $x=y$ appears in $C$ or $x \neq y$ appears in $C$ (thus they also appear in $C_1$ and $C_2$).  Let $Ax_i$ be a set of sentences in language ${\cal L}_i$ for $i\in\{1,2\}$.  Suppose there exists a model for $Ax_i \cup C_i$ for $i\in\{1,2\}$ and that these models have domains of same cardinality.  Then there exists a model for $C \cup Ax_1 \cup Ax_2$.
+**Theorem:** Let ${\cal L}$ be a finite language ${\cal L} = {\cal L}_1 \cup {\cal L}_2$ where ${\cal L}_1 \cap {\cal L}_2 = \emptyset$.  Let $C$ be a conjunction in language ${\cal L}$, and for $i\in \{1,2\}$ let $C_i$ be those literals from $C$ whose symbols appear only in ${\cal L}_i$, as well as equality and disequality literals.  Suppose further that if variables $x,y \in FV(C_1)\cap FV(C_2)$ then either $x=y$ appears in $C$ or $x \neq y$ appears in $C$ (thus they also appear in $C_1$ and $C_2$).  Let $Ax_i$ be a set of sentences in language ${\cal L}_i$ for $i\in\{1,2\}$.  Suppose there exists a model for $Ax_i \cup C_i$ for $i\in\{1,2\}$ and that **these models have domains of same cardinality**.  Then there exists a model for $C \cup Ax_1 \cup Ax_2$.