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Example 1: "John goes to work either by car [30'-40'], or by bus (at least 60'). Fred goes to work either by car [20'-30'], or in a carpool [40'-50']. Today John left home (t1) between 7:10 and 7:20, and Fred arrived (t4) at work between 8:00 and 8:10. We also know that John arrived (t2) at work about 10'-20' after Fred left home (t3)."
In this example, we have the disjunctive labeled constraints in Figure 1, where T0 represents the initial time (7:00) and where the granularity is in minutes. A label 'R0' is associated to elemental constraints belonging to constraints with only one disjunct. In constraints with more than one, mutually exclusive disjuncts, each disjunct is labeled with an exclusive label Rn (n>0). Thus,

- The label R0 is associated to "John left home between 7:10 and 7:20", "Fred arrived at work between 8:00 and 8:10", and "John arrived at work about 10'-20' after Fred left home". This is the *common context*.

- The label R1 is associated to "John goes by bus", and R2 to "John goes by car". - The label R3 is associated to "Fred goes in a carpool", and R4 to "Fred goes by car".
Definition 2 **(Inconsistent-Label-Sets).** An *Inconsistent-Label-Set* (I-L-Set) is a set of labels {labeli}
and represents a set of overall inconsistent elemental constraints. That is, they cannot all simultaneously hold. ◊
Theorem 1. Any label set that is a superset of an I-L-Set is also an I-L-Set. The proof is obvious. If a set of elemental constraints is inconsistent, any superset of it is also inconsistent. ◊
Definition 3. Elemental constraints {lecij.k} of an input disjunctive constraint lcij are pairwise disjoint. Thus, each 2-length set of labels from each pair of {lecij.k} is added to the set of I-L-Sets. That is, for each input constraint lcij ≡ {lecij.1, lecij.2, ..., lecij.l}, where lecij.k≡(ecij.k{labelij.k}) and |{labelij.k}|=1:
∀k,p∈(1,..,l) / k≠p, I-L-Sets ← I-L-Sets ∪ ({labelij.k}∪{labelij.p}) ◊
In the example of Figure 1, {R1 R2} and {R3 R4} are I-L-Sets. Other I-L-Sets existing in a labeled TCN will be detected in the reasoning processes later detailed in Section 4.