# Differences

This shows you the differences between two versions of the page.

 sav08:partial_congruences [2008/04/23 07:50]vkuncak 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 $i \ge 0$. - - **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: - $- \begin{array}{l} - ​\forall x_1,​\ldots,​x_n,​y_1,​\ldots,​y_n. \bigwedge_{i=1}^n (x_i,y_i) \in r\ \land \ f(x_1,​\ldots,​x_n) \in T \land f(y_1,​\ldots,​y_n) \in T \rightarrow \\ - (f(x_1,​\ldots,​x_n),​f(y_1,​\ldots,​y_n)) \in r -$