### 3.7.117. Hybrid-consistency

Denotes the fact that, for a given constraint involving both domain and set variables, there is a filtering algorithm that ensures hybrid -consistency. A constraint ctr defined on the distinct domain variables ${V}_{1}^{d},...,{V}_{n}^{d}$ and the distinct set variables ${V}_{n+1}^{s},...,{V}_{m}^{s}$ is hybrid -consistent if and only if:

• For every pair $\left({V}^{d},v\right)$ such that ${V}^{d}$ is a domain variable of ctr and $v\in \mathrm{𝑑𝑜𝑚}\left({V}^{d}\right)$, there exists at least one solution to ctr in which ${V}^{d}$ is assigned the value $v$.

• For every pair $\left({V}^{s},v\right)$ such that ${V}^{s}$ is a set variable of ctr, if $v\in \underline{{V}^{s}}$ then $v$ belongs to the set assigned to ${V}^{s}$ in all solutions to ctr and if $v\in \overline{{V}^{s}}\setminus \underline{{V}^{s}}$ then $v$ belongs to the set assigned to ${V}^{s}$ in at least one solution and is excluded from this set in at least one solution.