# Expanding a tuple via a monad or applicative

Suppose I have a term $$a :\alpha$$ of the Simply-Typed Lambda Calculus (in the following, $$\alpha, \beta, \gamma$$ stand for arbitrary types) and I want to lift it to a term

$$\lambda x_{\beta}. \;(x, \, a)$$

I could use a function $$\lambda z_{\alpha}, x. \;(x,\, z)$$.

Suppose we then form $$(b, a) : \beta \times \alpha$$, by applying $$\lambda x_{\beta}. \;(x, \, a)$$ to $$\,b_{\beta}$$.

We might want to add $$c$$ to the beginning of this to form $$(c, b, a) : \gamma \times \beta \times \alpha$$. We could do this (here $$\pi_1$$ and $$\pi_2$$ are projections)) by having a function $$\lambda z’_{\beta \times \alpha}, z. \,(z,\, \pi_1 z’,\, \pi_2 z’)$$. And again, we could cook up a function to form $$(d,\, c,\, b,\, a)$$ and $$(e,\,d,\, c,\, b,\, a)$$ (so on and so forth).

I could do things in the way above; however, I wondered whether there was a way of doing this kind of operation via an applicative or a monad. Then I could (ideally) use the operations of the monad or applicative, to lift term $$a$$ (perhaps into $$\lambda x.\,(x, \, a)$$, and then form these tuples $$(b, a), (c, b, a), (d, c, b, a)$$, etc, by the operations of the monad or applicative.

If you know a way of doing this I would be very interested.