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sav08:weak_monadic_logic_of_one_successor [2008/05/15 10:02] vkuncak |
sav08:weak_monadic_logic_of_one_successor [2008/05/15 10:35] vkuncak |
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| Note in particular that quantification is restricted to finite sets (elements of $D$). | Note in particular that quantification is restricted to finite sets (elements of $D$). | ||
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| Then we can define addition $N(Z) = N(X) + N(Y)$ by saying that there exists a set of carry bits $C$ such that the rules for binary addition hold: | Then we can define addition $N(Z) = N(X) + N(Y)$ by saying that there exists a set of carry bits $C$ such that the rules for binary addition hold: | ||
| \begin{equation*} | \begin{equation*} | ||
| - | \exists C.\ 0 \notin i\ \land \forall i.\ | + | \exists C.\ 0 \notin C\ \land \forall i.\ |
| \big(\begin{array}[t]{rcl} | \big(\begin{array}[t]{rcl} | ||
| ((i \in Z) &\leftrightarrow& ((i \in X) \oplus (i \in Y) \oplus (i \in C))\ \land\\ | ((i \in Z) &\leftrightarrow& ((i \in X) \oplus (i \in Y) \oplus (i \in C))\ \land\\ | ||