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Proving that the state of a controlled nonlinear system always stays inside a time moving bubble (or capture tube) amounts to proving the inconsistency of a set of nonlinear inequalities in the time-state space. In practice however, even with a good intuition, it is difficult for a human to find such a capture tube except for simple examples. In 2014, Jaulin et al. established properties that support a new interval approach for validating a quasi capture tube, i.e. a candidate tube (with a simple form) from which the mobile system can escape, but into which it enters again before a given time. A quasi capture tube is easy to find in practice for a controlled system. Merging the trajectories originated from the candidate tube yields the smallest capture tube enclosing it. This paper proposes an interval constraint programming solver dedicated to the quasi capture tube validation. The problem is viewed as a differential CSP where the functional variables correspond to the state variables of the system and the constraints define system trajectories that escape from the candidate tube "for ever". The solver performs a branch and contract procedure for computing the trajectories that escape from the candidate tube. If no solution is found, the quasi capture tube is validated and, as a side effect, a corrected smallest capture tube enclosing the quasi one is computed. The approach is experimentally validated on several examples having 2 to 5 degrees of freedom.