Trying to understand `filterM (const [True, False]) [1..5]`

filterMfuckery.hs

-- Given a list of numbers, generate a list of all possible sums.

-- Solution
map sum $ filterM (const [True, False]) [1..5]

-- Manual evaluation of `filterM (const [True, False]) [1..5]` using repeated substitution

filterM (const [True, False]) [1..5]

-- replace `filterM` with its definition
\p -> foldr (\x -> liftA2 (\flg -> if flg then (x:) else id) (p x)) (pure []) (const [True, False]) [1..5]

-- apply `(const [True, False])` to the lambda expression
foldr (\x -> liftA2 (\flg -> if flg then (x:) else id) ((const [True, False]) x)) (pure []) [1..5]

-- perform the first iteration using 1
	liftA2 (\flg -> if flg then (1:) else id) ((const [True, False]) 1) (pure [])
	liftA2 (\flg -> if flg then (1:) else id) [True, False] (pure [])
  
    -- liftA2 is defined as
		-- fmap f a <*> b
		fmap (\flg -> if flg then (1:) else id) [True, False] <*> (pure [])
    
    -- This is building a cartesian product, in this case with the empty list
    -- In this instance, the cartesian product would be every possible list, where I use the element from the first list
    -- (in this case the 1), and where I don't use it.
		[(1:), id] <*> (pure [])
		-- (<*>) is implemented for lists as follows
		[f x | f <- [(1:), id], x <- (pure [])]
		[[1], []]