Is it possible to build short proofs of arbitrary folds over a huge list?

Question

With the use of Merkle Trees, you can prove the presence of an element of a very big list, with an amount of information close to just logarithm of the size of the whole tree. Merkle proofs, thus, probabilistically confirm the result of this specific fold:

isMember :: ∀ t . List t -> Bool
isMember e = foldr (\ h t -> h == e || t) False

My question is if the same result can be extended to arbitrary folds; or, in other words, is it possible to prove arbitrary computations over a big list using some clever trick similar to Merkle Trees?

Formally

Given those 5 values:

1. H :: String -> Uint256, a fingerprint function
2. H(X) :: Uint256, the fingerprint of a dataset
3. F :: String -> String, an arbitrary computation
4. Y :: String, the claimed result of F(X)
5. A short proof

I need a verify function that returns true iff F(X) == Y, without having to run F(X); perhaps not even having the whole original dataset X.

Specific example

Suppose that you have a huge dataset (20 GB) distributed through a network of 100 computers that don't trust each-other. Is there any way for a computer to generate a claim such as the sum of this dataset is 1650874919052809!, so that other nodes are able to probabilistically verify that claim is correct, without actually running sum dataset?