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msol_over_strings [2007/05/27 12:43] vaibhav.rajan |
msol_over_strings [2007/05/27 13:27] vaibhav.rajan |
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[\![\exists v. F]\!]e &=& \exists S. S\ \mbox{is finite}\ \land\ S \subseteq N_0 \land [\![F]\!](e[v \mapsto S]) | [\![\exists v. F]\!]e &=& \exists S. S\ \mbox{is finite}\ \land\ S \subseteq N_0 \land [\![F]\!](e[v \mapsto S]) | ||
\end{eqnarray*} | \end{eqnarray*} | ||
+ | |||
+ | |||
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* Intersection: $(A = B \cap C) = (A \subseteq B \land A \subseteq C) \land (\forall A_1. A_1 \subseteq B \land A_1 \subseteq C \rightarrow A_1 \subseteq A)$ | * Intersection: $(A = B \cap C) = (A \subseteq B \land A \subseteq C) \land (\forall A_1. A_1 \subseteq B \land A_1 \subseteq C \rightarrow A_1 \subseteq A)$ | ||
* Union: $(A = B \cup C) = (B \subseteq A \land C \subseteq A) \land (\forall A_1. B \subseteq A_1 \land C \subseteq A_1 \rightarrow A \subseteq A_1)$ | * Union: $(A = B \cup C) = (B \subseteq A \land C \subseteq A) \land (\forall A_1. B \subseteq A_1 \land C \subseteq A_1 \rightarrow A \subseteq A_1)$ | ||
- | * Set difference: $(A = B \setminus C) = (A \cup C = B \land A \cap C = \emptyset)$ | + | * Set difference: $(A = B \setminus C) = (A \cup (B \cap C) = B \land A \cap C = \emptyset)$ |
(or just use element-wise definitions with singletons) | (or just use element-wise definitions with singletons) | ||
* If $k$ is a fixed constant, properties $\mbox{card}(A) \geq k$, $\mbox{card}(A)\leq k$, $\mbox{card}(A)=k$ | * If $k$ is a fixed constant, properties $\mbox{card}(A) \geq k$, $\mbox{card}(A)\leq k$, $\mbox{card}(A)=k$ | ||
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\end{equation*} | \end{equation*} | ||
+ | This does not give the smallest set containing both $u$ and $v$. The reflexive transitive closure, T is: | ||
+ | \begin{equation*} | ||
+ | (F \subseteq T )\land (\forall x. x \in S \rightarrow (x,x) \in T ) \land ((\exists k.(u,k) \in T \land (k,v) \in T) \rightarrow ((u,v) \in T)) | ||
+ | \end{equation*} | ||
+ | The underlying smallest set S containing $u$ and $v$ is given by: | ||
+ | \begin{equation*} | ||
+ | S = \{x | \exists k. ((k, x) \in T \lor (x,k) \in T) \} | ||
+ | \end{equation*} | ||
**Using transitive closure and successors:** | **Using transitive closure and successors:** | ||
* Constant zero: $(x=0) = One(x) \land \lnot (\exists y. One(y) \land s(y,x))$ | * Constant zero: $(x=0) = One(x) \land \lnot (\exists y. One(y) \land s(y,x))$ |