Explicit construction of benign subgroup for Higman's reversing operation

For the Higman reversing operation $\rho$ and for a set of integer-valued functions $\mathcal X$ the following has been proved. Let the subgroup $A_{\mathcal X}$ be benign in the free group $F$, let the respective finitely presented overgroup $K_{\mathcal X}$ with its finitely generated subgroup $L_{\mathcal X}$ be given for $A_{\mathcal X}$ explicitly, and let the set $\mathcal Y = \rho\mathcal X$ be obtained from $\mathcal X$ by the reversing operation $\rho$. Then $A_{\mathcal Y}$ also is benign in $F$ and, moreover, the finitely presented overgroup $K_{\mathcal Y}$ with its finitely generated subgroup $L_{\mathcal Y}$ can also be given explicitly for it. The current work is a step in development of a general algorithm for construction of explicit Higman embeddings of recursive groups into finitely presented groups.