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--- sav08:partial_congruences [2008/04/26 17:42]
damien
+++ sav08:partial_congruences [2015/04/21 17:30]
@@ 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.  We call these congruences partial congruences; they are simply congruences on a subset of the original set.
-**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 $i \ge 0$, by induction.
-  * $i = 0$: s = s
-  * $i \rightarrow i+1$:
-$C^{i+1}(s) \cap T^2 = C(C^i(s) \cap T^2) \cap T^2 $, by induction hypothesis $C^i(s) \cap T^2 =s$.
-Thus $C(C^i(s) \cap T^2) \cap T^2 = C(s) \cap T^2$.
-As $C$ adds only needed term for congruence, the added term are either not in $T^2$ or $s$ in not a congruence.
-By hypothesis $s$ is a congruence, so $C(s) \cap T^2 = s$.
-Therefore $C^{i+1}(s) \cap T^2 = s$.
-**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:
-\[
-\begin{array}{l}
-   \forall x_1,\ldots,x_n,y_1,\ldots,y_n. \bigwedge_{i=1}^n (x_i,y_i) \in r\ \land \ f(x_1,\ldots,x_n) \in T \land f(y_1,\ldots,y_n) \in T  \rightarrow \\
-    (f(x_1,\ldots,x_n),f(y_1,\ldots,y_n)) \in r
-\]