# Comparison chain

### From HaskellWiki

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(Category:Mathematics) |
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</haskell> | </haskell> | ||

+ | * You can represent a successful sequence of comparisons with <code>Just a</code> and a failed sequence with <code>Nothing</code>: | ||

+ | |||

+ | <haskell> | ||

+ | 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] | ||

+ | </haskell> | ||

[[Category:Mathematics]] | [[Category:Mathematics]] |

## Revision as of 19:58, 27 October 2010

## Contents |

## 1 Problem

Question:

The compiler doesn't accepta <= x <= b

Answer: The expression cannot be parsed,

because the infix symbol <=

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.)

## 2 Solutions

### 2.1 simple

- You must be aware, that the mathematical notation is shorthand for . Consequently a possible Haskell solution is .a <= x && x <= b
- Another fine mathematical notation is . 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 (), orisInRange a b x

(<?) :: Ord a => a -> (a,a) -> Bool (<?) = flip (uncurry isInRange)

- (). In case of integers you can use thex <? (a,b)function frominRangeclass.Ix

- You can easily write a function, which checks if a list of numbers increases monotonicly.

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]

### 2.2 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]