# Dynamic Controllability of Processes without Surprises
## 6 Transforming Sudden Start into Sudden Termination
Then all valid scenarios \(\sigma^{\prime}\) for \(P^{\prime}\) there is a valid scenario \(\sigma\) for \(P\)._  
Proof. : Let \(S\) be any activity requiring advance notice \(k(S)\) in \(P\). Let \(wait_{5}\) be the waiting task associated to \(S\) in \(P^{\prime}\). Let \(\Gamma\) be the set of constraints on \(S.s\) in \(P\); let \(\Gamma^{\prime}\) be the set of corresponding constraints on \(wait_{5}.e\) in \(P^{\prime}\). Let \(\sigma^{\prime}\) be a valid scenario for \(P^{\prime}\). Let \(\sigma\) be the projection of \(\sigma^{\prime}\) over the elements of \(P\). From \(ubc(wait_{5}.e,S.s,0),lbc(wait_{5}.e,S.s,0)\in\mathcal{C}^{\prime}\), and \(\sigma^{\prime}\) valid, then it follows that \(\overline{wait_{5}.e}=\overline{S.s}\). Since \(\overline{wait_{5}.e}=\overline{S.s}\), and \(\sigma^{\prime}\) is valid, it follows that:  
* \(\forall ubc(a,wait_{5}.e,\delta)\): \(\overline{S.s}\leq\overline{a}+\delta\);
* \(\forall lbc(wait_{5}.e,b,\delta)\): \(\overline{b}\leq\overline{S.s}+\delta\);
* \(\forall lbc(a,wait_{5}.e,b,\delta)\): \(\overline{b}\leq\overline{S.s}+\delta\). which means that \(\Gamma^{\prime}\) satisfied by \(\sigma^{\prime}\) in \(P^{\prime}\Rightarrow\Gamma\) satisfied by \(\sigma\) in \(P\). From \(d(wait_{5},k(S),h,sc),ubc(wait_{5}.e,S.s,0),lbc(wait_{5}.e,S.s,0)\in\mathcal{C}^{\prime}\), and \(\sigma^{\prime}\) valid, it follows that \(\overline{S.s}\geq\overline{wait_{5}}+k(S)\) in \(P^{\prime}\). So \(\exists\overline{k_{5}}\in\sigma\) with \(\overline{k_{5}}=\overline{wait_{5}}:\overline{k_{5}}+k(S)\leq\overline{S.s}\) in \(P\). All other process elements of \(P\) are the same in \(P^{\prime}\). Thus, all other time point assignments of \(\sigma^{\prime}\) that meet constraints in \(P^{\prime}\), correspond to time point assignments of \(\sigma\) that meet constraints in \(P\). So \(\sigma\) is a valid scenario for \(P\). **Lemma 2** (Completeness). : _Let \(P(N,E,\mathcal{C},k,\Omega)\) be a process model; let \(\Psi(P)=P^{\prime}(N^{\prime},E,\mathcal{C}^{\prime},\)\(k^{\prime},\Omega)\) be a process model obtained by applying the transformation of Definition 13 to \(P\). Then for all valid scenarios \(\sigma\) for \(P\) there exists a valid scenario \(\sigma^{\prime}\) for \(P^{\prime}\)._Proof. : The proof can be constructed like the proof of soundness, with the difference that the timestamp for each advance notice \(\overline{k_{S}}\) is assigned to the start event of the waiting task \(wait_{S}\). Figure 10 shows the process model resulting from the application of the transformation Definition 13 to the process model of Figure 3.  
We can now show that we can treat the sudden start of an activity in a process model \(P\) as the sudden termination of its associated waiting task in the transformed process model \(P^{\prime}\). **Theorem 13** (Sudden Start as Sudden Termination of a Waiting Task). : _Let \(P(N,E,C,k,\Omega)\) be a process model; let \(P^{\prime}(N^{\prime},E,C^{\prime},k^{\prime},\Omega)\) be a process model obtained by applying the transformation of Definition 13 to \(P\). Let \(S\) and \(C\) be activities in \(P\), with \(d(C,C.d_{min},C.d_{max},c)\) and \(k(S)>0\). Let \(wait_{S}\) be the waiting task associated to \(S\) in \(P^{\prime}\). Then \(SSP_{S,C}\) in \(P\Rightarrow STP_{wait_{S},C}\) in \(P^{\prime}\)._  
Proof. : The waiting task is a semi-contingent activity in an STP and corresponds to the precedence constraint of \(S\). From \(SSP_{S,C}\) and Definition 5 it follows that \(\forall S.s,w:S.s-w\geq k(S),\ \forall C.s\ \exists C.d_{min}\leq C.d\leq C.d_{max}: \ \
eg\Phi_{P}\), where \(\Phi_{P}\) is the set of constraints in \(P\). From the transformation, \(wait_{S}\) is the semi-contingent waiting task associated to \(S\) in \(P^{\prime}\). All temporal constraints on \(S.s\) are bound to \(wait_{S}.e\) in \(P^{\prime}\). So \(\forall wait_{S}.e,w:wait_{S}.e-w\geq k(S),\ \forall C.s\exists C.d_{min} \leq C.d\leq C.d_{max}:\ \
eg\Phi^{\prime}_{P}\), where \(\Phi^{\prime}_{P}\) is the set of constraints in \(P^{\prime}\).