Memoization for subsequent fold calls

I'm looking for a technique that allows for memoization between subsequent fold calls against the lists that is being prepended.

I looked at memoize library but this doesn't seem to support memoization of higher-order functions, which is the case for folds.

I also tried the technique with lazy evaluated map of results but to no avail.

Here's simple example code:

module Main where

import Data.Time

printAndMeasureTime :: Show a => a -> IO ()
printAndMeasureTime a = do
  startTime <- getCurrentTime
  print a
  stopTime <- getCurrentTime
  putStrLn $ " in " ++ show (diffUTCTime stopTime startTime)

main = do
  let as = replicate 10000000 1
  printAndMeasureTime $ foldr (-) 0 as -- just to resolve thunks
  printAndMeasureTime $ sum as
  printAndMeasureTime $ sum (1:as) -- recomputed from scratch, could it reuse previous computation result?
  printAndMeasureTime $ length (as)
  printAndMeasureTime $ length (1:as) -- recomputed from scratch, could it reuse previous computation result?

and the output:

0
 in 1.125098223s
10000000
 in 0.096558168s
10000001
 in 0.104047058s
10000000
 in 0.037727126s
10000001
 in 0.041266456s

Times suggest that folds are computed from scratch. Is there a way to make the subsequent folds reuse previous fold results?

Solution

Make a data type!

module List (List, _elements, _sum, _length, toList, cons) where

data List = List
  { _elements :: [Int]
  , _sum :: !Int
  , _length :: !Int
  }

toList :: [Int] -> List
toList xs = List xs (sum xs) (length xs)

cons :: Int -> List -> List
cons x (List xs t n) = List (x:xs) (x+t) (1+n)

Note that the List type is exported, but the List constructor is not, so that the only way to construct a List is using the toList function (commonly called a "smart constructor").