[Haskell-cafe] Is it possible to change the environment (reader) in applicative style?

[Rodrigo Stevaux]

It is easy to read an environment in applicative style, ie:

type Env = [(String, Int)]
data Term = Add Term Term | Number Int | Var String deriving Show
eval :: Term -> Env -> Int
eval (Add a b) = (+) <$> eval a <*> eval b
eval (Var name) = fetch name
eval (Number i) = pure i

fetch :: String -> Env -> Int
fetch name = fromJust . lookup name

But can the eval function change the Env being passed, as to implement
a "let" operation, without using monads? I tried I lot but ultimately
I resorted to (>>=) in the function monad:

bind f k = \r -> k (f r) r

I do not think so, because in applicative style each operand can have
an effect (reading the environment) but can not affect other operands
(including the next ones), i.e., there is no notion of sequencing in
applicatives

Is this reasoning right?