Type inhabitation

In type theory, a branch of mathematical logic, in a given typed calculus, the type inhabitation problem for this calculus is the following problem:^[1] given a type ${\displaystyle \tau }$ and a typing environment ${\displaystyle \Gamma }$, does there exist a ${\displaystyle \lambda }$-term M such that ${\displaystyle \Gamma \vdash M:\tau }$? With an empty type environment, such an M is said to be an inhabitant of ${\displaystyle \tau }$.

Relationship to logic

In the case of

Girard's paradox shows that type inhabitation is strongly related to the consistency of a type system with Curry–Howard correspondence. To be sound, such a system must have uninhabited types.

Formal properties

For most typed calculi, the type inhabitation problem is very

${\displaystyle \Gamma \!\vdash }$

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