### 3.7.92. Entailment

Denotes the fact that the catalogue mentions a sufficient condition for the entailment of a constraint. Consider a constraint $𝒞\left({V}_{1},{V}_{2},...,{V}_{n}\right)$ and the potential sets of values $\mathrm{𝑑𝑜𝑚}\left({V}_{1}\right),\mathrm{𝑑𝑜𝑚}\left({V}_{2}\right),...,\mathrm{𝑑𝑜𝑚}\left({V}_{n}\right)$ that can respectively be assigned to the domain variables ${V}_{1},{V}_{2},...,{V}_{n}$. The constraint $𝒞\left({V}_{1},{V}_{2},...,{V}_{n}\right)$ is entailed if and only if $𝒞\left({V}_{1},{V}_{2},...,{V}_{n}\right)$ holds whatever values ${\mathrm{𝑣𝑎𝑙}}_{1}\in \mathrm{𝑑𝑜𝑚}\left({V}_{1}\right),{\mathrm{𝑣𝑎𝑙}}_{2}\in \mathrm{𝑑𝑜𝑚}\left({V}_{2}\right),...,{\mathrm{𝑣𝑎𝑙}}_{n}\in \mathrm{𝑑𝑜𝑚}\left({V}_{n}\right)$ will respectively be assigned variables ${V}_{1},{V}_{2},...,{V}_{n}$.

Entailment is usually not considered as very important when designing a filtering algorithm, even if it can sometimes save waking again and again a constraint that will for sure be satisfied. From a modelling point of view, entailment detection is mandatory for coming up with the reified version of a constraint (see also reified automaton constraint).