Updating (ni c'ij nj) ;c'ij≡{ec'ij., ec'ij.2, ..., ec'ij.l*}, a disjunctive metric constraint.*

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The function *Put-Labels(c'*ij) returns a labeled-constraint lc'ij≡{lec'ij.1, lec'ij.2, ..., lec'ij.l},
associating an exclusive label to each elemental constraint in c'ij. If there is only one disjunct in c'ij, the label in the unique elemental constraint is {R0}. Otherwise, each pair of labels in lc'ij is added to the set of I-L-Sets, since elemental constraints in c'ij are pairwise disjoint (*Definition 3*). By using the Inverse function on non-labeled constraints, the *Inverse*lc function is:
Inverselc ({(ecij.k{labelij.k})}) =def {(Inverse (ecij.k) {labelij.k})}
The described updating process is performed *each time* that one new input constraint c'ij is asserted on a previous TCN. Thus, an initial TCN with no nodes, no constraints, and no I-L-Sets is assumed (Figure 9). At each new input constraint (c'ij), the TCN is incrementally updated and closured. That is, if c'ij is consistent (*Consistency-Test* function), the constraint c'ij is added to the TCN, the closure process (*Closure* function) propagates its effects to all TCN, and the new TCN is obtained. A new updating process can be performed on this new TCN, and so on successively.

## 4.1.1. The Consistency-Test Function

The Consistency-Test function (Figure 3) is based on the operation ⊕lc. A new input constraint lc'ij between nodes ni and nj is consistent if it temporally intersects with the previously existing constraint lcij between these nodes. Moreover, the Consistency-Test function can detect new I-L-Sets:

i) If the new constraint lc'ij is consistent with the existing constraint lcij, and two elemental constraints ecij.p∈lc'ij, ecij.k∈lcij do not intersect (ecij.k ⊕ ecij.p=∅), then the label set
{labelij.k}∪{labelij.p} is an I-L-Set and should be added to the current set of I-L-Sets.

ii) If an existing elemental constraint between nodes ni and nj (lecij.k∈lcij) does not intersect with the new constraint lc'ij, then {labelij.k} is an I-L-Set and should be added to the current set of I-L-Sets.