Step by Step solution about How to resolve the algorithm Compare a list of strings step by step in the Haskell programming language

The provided code defines four functions that perform different checks on a given list. Let's break down each function in detail:

1. allEqual:

• The allEqual function takes a list of elements of type a (assuming a is an instance of the Eq typeclass) as input and returns a Bool value indicating whether all elements in the list are equal.

• It achieves this by using zipWith, which pairs up corresponding elements from two lists, and then applies the == operator to each pair to check if they are equal. The results are collected into a list of Bool values.

• Finally, it uses and to combine all the Bool values, resulting in True if all elements are equal and False otherwise.

2. allIncr:

• The allIncr function operates on a list of elements of type a, where a must implement the Ord typeclass (which allows for ordering comparisons). It returns a Bool value indicating whether all elements in the list are strictly increasing (i.e., each element is greater than the previous one).

• Similar to allEqual, it employs zipWith to pair up consecutive elements in the list and checks if each pair satisfies the < condition. The results are stored in a list of Bool values.

• Finally, it uses and to combine the Bool values to determine whether all elements in the list are strictly increasing.

3. allEq:

• The allEq function, like allEqual, takes a list of elements of type a where a is an instance of Eq. It checks if all elements in the list are equal. However, it uses a different approach compared to allEqual.

• It uses until to iteratively check if each element in the list is equal to the head (first element) of the list. The iteration stops when either the list is empty or when an element is found that is not equal to the head.

• If the iteration finishes without finding any unequal elements, it returns True. Otherwise, it returns False.

4. allInc:

• The allInc function is similar to allIncr, but instead of checking for strict increasing order, it checks for non-decreasing order (i.e., each element is greater than or equal to the previous one).

• It also utilizes the until function to iteratively check if each element is greater than or equal to the head of the list, stopping when the list is empty or when an element is found that violates this condition.

• As with allEq, if the iteration completes without finding any violations, it returns True, and if violations are found, it returns False.

allEqual :: Eq a => [a] -> Bool
allEqual xs = and $ zipWith (==) xs (tail xs)
 
allIncr :: Ord a => [a] -> Bool
allIncr xs = and $ zipWith (<) xs (tail xs)


allEqual
  :: Eq a
  => [a] -> Bool
allEqual [] = True
allEqual (h:t) = foldl (\a x -> a && x == h) True t

allIncreasing
  :: Ord a
  => [a] -> Bool
allIncreasing [] = True
allIncreasing (h:t) = fst $ foldl (\(a, x) y -> (a && x < y, y)) (True, h) t


allEq
  :: Eq a
  => [a] -> Bool
allEq [] = True
allEq (h:t) =
  null . snd $ 
  until 
    (\(x, xs) -> null xs || x /= head xs)
    (\(_, x:xs) -> (x, xs)) 
    (h, t)

allInc
  :: Ord a
  => [a] -> Bool
allInc [] = True
allInc (h:t) =
  null . snd $
  until
    (\(x, xs) -> null xs || x >= head xs)
    (\(_, x:xs) -> (x, xs))
    (h, t)