Critical Pair
Let 
and 
be two rules
of a term rewriting system, and suppose these
rules have no variables in common. If they do, rename the variables. If 
is a subterm
of 
(or the term 
itself) such that
it is not a variable, and the pair 
is unifiable
with the most general unifier 
, then 
and the result of replacing 
in 
by 
are called
a critical pair.
The fact that all critical pairs of a term rewriting system are joinable, i.e., can be reduced to the same expression, implies that the system is locally confluent.
For instance, if 
and 
,
then 
and 
would form
a critical pair because they can both be derived from 
.
Note that it is possible for a critical pair to be produced by one rule, used in two different ways. For instance, in the string rewrite "AA" -> "B", the critical pair ("BA", "AB") results from applying the one rule to "AAA" in two different ways.
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