# Comparison chain

## Problem

Question: The compiler doesn't accept a <= x <= b. Why?

Answer: The expression cannot be parsed, because the infix symbol <= has no (left or right) associativity.

In languages like C the expression is parsed as (a <= x) <= b which is even worse. The first part is evaluated to a boolean value, which is then compared with b. (For C "boolean" and "integer" are the same type.)

## Solutions

### simple

• You must be aware, that the mathematical notation $a \le x \le b$ is shorthand for $a \le x\ \land\ x \le b$. Consequently a possible Haskell solution is a <= x && x <= b.
• Another fine mathematical notation is $x \in [a,b]$. You can roll your own function
isInRange :: Ord a => a -> a -> a -> Bool
isInRange lower upper x = lower <= x && x <= upper

to capture this notation (isInRange a b x), or
(<?) :: Ord a => a -> (a,a) -> Bool
(<?) = flip (uncurry isInRange)

(x <? (a,b)). In case of integers you can use the inRange function from Ix class.
• You can easily write a function, which checks if a list of numbers increases monotonically.
monotonicIncreasing :: Ord a => [a] -> Bool
monotonicIncreasing xs = and (zipWith (<=) xs (tail xs))

You can use that for the initial problem by monotonicIncreasing [a,x,b].

### complex

• For more complex checks of whether an element is contained in some ranges, you should have a look at the ranged sets library.
• If you want to program more complex chains with different kinds of comparisons, try the following code.
module ChainRelation where

{- * chains of relations (comparison, subsets, logical implications etc.) -}

infixr 4 &-, -&

type Rel   a = (a -> a -> Bool)
type Chain a = [(Rel a, a)]

endChain :: Chain a
endChain = []

-- separate comparison and operand
(&-) :: Rel a -> (a, Chain a) -> Chain a
rel &- (x,xs) = (rel,x):xs

-- separate operand and comparison
(-&) :: a -> Chain a -> (a, Chain a)
(-&) = (,)

-- check if all comparisons are true
check :: (a, Chain a) -> Bool
check (x,chain) =
let (rels,xs) = unzip chain
in  and (zipWith3 id rels (x:xs) xs)

example1 :: Bool
example1 =
check (1 -& (<) &- 5 -& (==) &- 5 -& (<=) &- 10 -&
(endChain :: Chain Integer))

{- * specialised infix operators for comparison -}

infixr 4 ==:, /=:, <:, >:, <=:, >=:

(==:), (/=:), (<:), (>:), (<=:), (>=:) :: Ord a =>
a -> (a, Chain a) -> (a, Chain a)
(==:) = lift (==)
(/=:) = lift (/=)
(<:)  = lift (<)
(>:)  = lift (>)
(<=:) = lift (<=)
(>=:) = lift (>=)

lift :: Rel a -> a -> (a, Chain a) -> (a, Chain a)
lift f x (y,chain) = (x, (f,y):chain)

example2 :: Bool
example2 =
check (1 <: 5 ==: 5 <=: 10
-& (endChain :: Chain Integer))

• You can represent a successful sequence of comparisons with Just a and a failed sequence with Nothing:
module ChainRelation2 where

import Data.Maybe (isJust)

lift :: (a -> b -> Bool) -> Maybe a -> b -> Maybe b
lift f (Just x) y | f x y = Just y
lift _ _ _ = Nothing

(<:)  = lift (<)
(<=:) = lift (<=)
(*:)  = lift elem

example = isJust (Just 4 <: 5 <=: 6 *: [6,7])

-- Python equivalent: 4 < 5 <= 6 in [6,7]