\(\hat{\mathcal {R}}_{k} = \langle \hat{c}_{k}, \hat{G}_{k}^*  \rangle \) 

\(\tilde{\mathcal {R}}_{k} = \textit {reduce}(\tilde{\mathcal {R}}_{k})\)  using [1]}

\(\hat{\mathcal{R}}_{k} = \langle \hat{c}_{k}, \hat{G}_{k}^* \rangle\)



\(\tilde{\mathcal{R}}_{k} = \textit{reduce}(\tilde{\mathcal{R}}_{k})\) using [1]