mjhjmtu

In the simply typed lambda calculus, one can show the following result, known as "preservation under substitution":

• If $Gamma vdash v : tau_1$ and $(x : tau_1) vdash t : tau_2$,
  then $Gamma vdash [v/x]t : tau_2 $.

However, the proof of this relies on the property of permutation, that we can rearrange contexts and it will preserve typing of terms.

I'm wondering, can we prove a similar property for dependently typed languages? The problem is that, here, permutation may not hold, since telescopes are used in place of environments, and the types themselves may refer to variables. Moreover, since the types and terms overlap, we have to substitute in $Gamma$ and $tau_2$ as well.

Does anyone have a good reference to proving such a preservation property for a dependently typed language? Are there tricks that are used to avoid the permutations?

The property, which I would call "typing of substitution" should hold in any type theory, and is not dependent on the exchange property (which I assume is what you mean by permutation)

The key is that you need to generalize the inductive hypothesis to when the variable in t appears in a context. So for a dependent type theory you prove

• If $Gamma vdash t_1 : tau_1$ and $Gamma, x : tau_1, Delta vdash t_2 : tau_2$ then $Gamma,Delta[t_1/x] vdash t_2[t_1/x] : tau_2[t_1/x]$