instantiations {(t1=0), (t2=0), (t3=0)} are consistent with the existing constraints involved among (T0, t1, t2, t3), but this partial solution cannot be extended to the overall TCN.

![0_image_0.png](0_image_0.png)

Let's consider the TCN with labeled constraints. For reasons of simplicity, we only denote the labeled constraints among (T0, t1, t2, t3):
(T0 {[5 45]{R0 R5}, [0 45]{R0 R4}, [0 45]{R0 R3}} t1),
(T0 {[0 25]{R2 R0}, [5 60]{R1 R0 R4}, [25 60]{R1 R0 R5}, [0 60]{R1 R0 R3}} t2),
(T0 {[25 55]{R0 R2 R3}, [0 15]{R0 R5}, [0 35]{R0 R1 R4}, [5 55]{R0 R1 R3}, [5 35]{R0 R2 R4}} t3),
(t1 {([-5 35]{R0 R2}, [-40 5]{R0 R1}} t2),
(t1 {[-15 15]{R0 R4}, [-35 -5]{R0 R3}, [5 35]{R0 R5}} t3),
(t2 {[5 30]{R1 R0 R4}, [-45 -25]{R2 R0 R3}, [25 50]{R1 R0 R5}, [-15 10]{R1 R0 R3}, [-25 -5]{R2 R0 R4}, [-5 15]{R2 R0 R5}} t3).

The set of I-L-Sets is {{R1 R2} {R3 R4} {R3 R5} {R4 R5}}. From this labeled TCN and the set of I-L-Sets, we can deduce that instantiations {(t1=0), (t2=0), (t3=0)} are not overall consistent. These instantiations are not *locally* consistent with the *labeled constraints* in the subTCN (T0, t1, t2, t3): All label sets associated to possible simultaneous fulfillment of
(T0 {[0 0]} t1), (T0 {[0 0]} t2) and (T0 {[0 0]} t3)
are I-L-Sets. That is, all label sets in the Cartesian product
{{R0 R4} {R0 R3}} Χ {{R2 R0} {R1 R0 R3}} Χ {{R0 R5} {R0 R1 R4}}
are I-L-Sets. Thus, the set of I-L-Sets can be used to deduce consistency of a set of labeled elemental constraints and to obtain a *globally consistent* labeled-TCN.

Theorem 9. Let's assume a labeled-TCN of n nodes (and the corresponding complete and sound set of I-L-Sets) and a local set of k (1≤k≤( 2 n)) labeled elemental constraints in the TCN, each one of which is between any pair of nodes:
{lec1, lec2,....., leck} ≡ {(ec1 {label1}), (ec2 {label2}), ..., (eck {labelk})}.