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--- sav08:partial_congruences [2008/04/23 07:43]
vkuncak created
+++ sav08:partial_congruences [2008/04/23 07:49]
vkuncak
@@ Line 1: / Line 1: @@
 ====== Partial Congruences ======
-Instead of a congruence on the (typically infinite) set of all ground terms we will compute congruences on a given finite set of ground terms.
+Instead of a congruence on the (typically infinite) set of all ground terms we will compute congruences on a given finite set of ground terms.  We call these congruences partial congruences; they are simply congruences on a subset of the original set.
-Let $T$ be a finite set of ground terms.
+**Theorem:** Let $T$ be a set of ground terms and $s$ a congruence on $T$.  Then $CC(s) \cap T^2 = s$ where $CC(s)$ denotes the congruence closure of $s$.
+**Proof:**
+Show $C^i(s) \cap T^2 = s$ for all $i \ge 0$.
+**Proof End.**
+When checking a formula we compute congruences on a finite set of terms that occur in the formula.
+We apply the congruence condition only to terms that already exist in the set, using congruence condition:
+\[
+   \forall x_1,\ldots,x_n,y_1,\ldots,y_n. \bigwedge_{i=1}^n (x_i,y_i) \in r \land f(x_1,\ldots,x_n) \in T \land f(y_1,\ldots,y_n) \in T  \rightarrow (f(x_1,\ldots,x_n),f(y_1,\ldots,y_n)) \in r
+\]