### 3.7.92. Entailment

Denotes the fact that the catalogue mentions a sufficient condition for the entailment of a constraint. Consider a constraint $\mathrm{\pi }\left({V}_{1},{V}_{2},...,{V}_{n}\right)$ and the potential sets of values $\mathrm{\pi \pi \pi }\left({V}_{1}\right),\mathrm{\pi \pi \pi }\left({V}_{2}\right),...,\mathrm{\pi \pi \pi }\left({V}_{n}\right)$ that can respectively be assigned to the domain variables ${V}_{1},{V}_{2},...,{V}_{n}$. The constraint $\mathrm{\pi }\left({V}_{1},{V}_{2},...,{V}_{n}\right)$ is entailed if and only if $\mathrm{\pi }\left({V}_{1},{V}_{2},...,{V}_{n}\right)$ holds whatever values ${\mathrm{\pi £\pi \pi }}_{1}\beta \mathrm{\pi \pi \pi }\left({V}_{1}\right),{\mathrm{\pi £\pi \pi }}_{2}\beta \mathrm{\pi \pi \pi }\left({V}_{2}\right),...,{\mathrm{\pi £\pi \pi }}_{n}\beta \mathrm{\pi \pi \pi }\left({V}_{n}\right)$ 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 alsoΒ reified automaton constraint).