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In the formulation of type theory I'm reading, (x:A) is an expression of type A. This would seem to imply ((x:A):A) and (((x:A):A):A)... Is this a common feature of type theories? Or am I reading too much into an ambiguous sentence?

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My guess would be that $(x:A)$ is only assumed to be an expression of type $A$ when $x$ is a variable. Therefore, since $(x:A)$ is not a variable, probably $((x:A):A)$ is not well-formed. – goblin Jun 17 '14 at 19:20
up vote 0 down vote accepted

This depends on the type-system, nevertheless, it is a common feature. However, in some type-systems it is possible to write something like:

$$(\lambda x.\ x) : (\alpha \to \mathtt{int})$$ (note that here the most general type for $\lambda x.\ x$ is $\alpha \to \alpha$) which would result in a value of type $\mathtt{int} \to \mathtt{int}$.

I hope this helps ;-)

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