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--- sav08:interpretation_quotient_under_congruence [2008/04/02 22:09]
vkuncak
+++ sav08:interpretation_quotient_under_congruence [2008/04/02 22:36]
vkuncak
@@ Line 72: / Line 72: @@
 \]
-**Lemma 1:** $I_Q = ([D],I_Q)$ is a well-defined interpretation, that is,
+**Lemma 0:** For all $x_1,\ldots,x_n \in D$,
-  * $[D]$ is non-empty;
+\[
-  * for each function symbol $f$ with $ar(f)=n$, the relation $\alpha_Q(f)$ is a total function $[D]^n \to [D]$.
+    ([x_1],\ldots,[x_n]) \in \alpha_Q(R) \mbox{ iff }  (x_1,\ldots,x_n) \in \alpha(R)
+\]
+**Lemma 1:** For each function symbol $f$ with $ar(f)=n$, the relation $\alpha_Q(f)$ is a total function $[D]^n \to [D]$ and for all $x_1,\ldots,x_n \in D$,
+\[
+    \alpha_Q(f)([x_1],\ldots,[x_n]) = [\alpha(f)(x_1,\ldots,x_n)]
+\]
 **Lemma 2:** For each term $t$ we have $e_T(t)(I_Q) = [e_T(t)(I)]$.