I need to define a haskell function:
func :: Int -> Int
func 1 = 1
func 2 = 2
func x = x+1
So that it allows only positive numbers. I've already had a look at a similar question: Non-negative integers
And wrote this:
newtype Positive a = Positive a
toPositive :: (Num a, Ord a) => a -> Positive a
toPositive x
| x < 0 = error "number cannot be negative"
| otherwise = Positive x
func :: Positive a -> a
func (Positive n) = n
Which is however already throwing errors. Thoughts?
Update:
Sample error:
*Main> func 1
<interactive>:32:6:
No instance for (Num (Positive a0)) arising from the literal `1'
Possible fix: add an instance declaration for (Num (Positive a0))
In the first argument of `func', namely `1'
In the expression: func 1
In an equation for `it': it = func 1
*Main>
You forgot to call toPositive to convert an Int to a Positive. Call it this way:
func $ toPositive 1
Also, a quirk of Haskell is its handling of negative number literals. To avoid confusion with the subtraction operator, you must wrap them in parentheses:
func $ toPositive (-1)
Related
``
I'm a Haskell newbie and I'm defining a function that given an Int n it tells if a number is prime or not by searching for an 2<=m<=sqrt(n) that mod n m ==0
if such m exists, then n is not prime, if not then n is prime.
I'm trying to define a list with numbers m between 2 and sqrt n, that mod n m ==0
My thought is that if the list is empty then n is prime, it not, n is not prime
`
isprime' :: Int -> Bool
isprime' n | l == [] = True
| otherwise = False
where
l = [x|x<-[2.. sqrt n], mod n x == 0]
`
But there seems to be a problem with sqrt n when I run my code and I can't understand it. can someone explain what I'm doing wrong/what to change for my code to run/ and why there's an error?
Running the code gives the following error
test.hs:9:28: error:
• No instance for (Floating Int) arising from a use of ‘sqrt’
• In the expression: sqrt n
In the expression: [2 .. sqrt n]
In a stmt of a list comprehension: x <- [2 .. sqrt n]
|
9 | l = [x|x<-[2.. sqrt n], mod n x == 0]
| ^^^^^^
You are correct in saying that the error is with sqrt, but the rest is pretty opaque to a new Haskell developer. Lets try by checking the type of sqrt to see if that helps.
Prelude> :t sqrt
sqrt :: Floating a => a -> a
Here I'm using ghci to interactively write code. :t asks for the type of the preceeding expression. The line sqrt :: Floating a => a -> a says sqrt takes in some floating point number a and returns something of the same type.
Similar to our error message we see this Floating thing. this thing is a typeclass but for the sake of solving this problem we'll save understanding those for later. In essence, haskell is trying to tell you that Int is not floating point number which is what sqrt expects. We can amend that by turning our Int into a Float with fromIntegral which is a really general function for turning number types into one another. (see Get sqrt from Int in Haskell)
isprime' :: Int -> Bool
isprime' n | l == [] = True
| otherwise = False
where
asFloat :: Float -- new! - tell fromIntegral we want a float
asFloat = fromIntegral n -- new! turn n into a Float
l = [x|x<-[2..sqrt asFloat], mod n x == 0]
This also errors! but it's a new one!
test.hs:10:48: error:
• Couldn't match expected type ‘Int’ with actual type ‘Float’
• In the second argument of ‘mod’, namely ‘x’
In the first argument of ‘(==)’, namely ‘mod n x’
In the expression: mod n x == 0
|
10 | l = [x|x<-[2..sqrt asFloat], mod n x == 0]
| ^
this is saying that x is suddenly a Float. When we changed to [2..sqrt asFloat] we now have made a list of Floats ([Float]). We need to change that back to [Int]. we can do that by calling floor on the result of the square root.
isprime' :: Int -> Bool
isprime' n | l == [] = True
| otherwise = False
where
asFloat :: Float
asFloat = fromIntegral n
l = [x|x<-[2..floor (sqrt asFloat)], mod n x == 0] -- new! I added floor here to change the result of sqrt from a `Float` into a `Int`
This now correctly compiles.
I wrote my first program in Haskell today. It compiles and runs successfully. And since it is not a typical "Hello World" program, it in fact does much more than that, so please congrats me :D
Anyway, I've few doubts regarding my code, and the syntax in Haskell.
Problem:
My program reads an integer N from the standard input and then, for each integer i in the range [1,N], it prints whether i is a prime number or not. Currently it doesn't check for input error. :-)
Solution: (also doubts/questions)
To solve the problem, I wrote this function to test primality of an integer:
is_prime :: Integer -> Bool
is_prime n = helper n 2
where
helper :: Integer -> Integer -> Bool
helper n i
| n < 2 * i = True
| mod n i > 0 = helper n (i+1)
| otherwise = False
It works great. But my doubt is that the first line is a result of many hit-and-trials, as what I read in this tutorial didn't work, and gave this error (I suppose this is an error, though it doesn't say so):
prime.hs:9:13:
Type constructor `Integer' used as a class
In the type signature for `is_prime':
is_prime :: Integer a => a -> Bool
According to the tutorial (which is a nicely-written tutorial, by the way), the first line should be: (the tutorial says (Integral a) => a -> String, so I thought (Integer a) => a -> Bool should work as well.)
is_prime :: (Integer a) => a -> Bool
which doesn't work, and gives the above posted error (?).
And why does it not work? What is the difference between this line (which doesn't work) and the line (which works)?
Also, what is the idiomatic way to loop through 1 to N? I'm not completely satisfied with the loop in my code. Please suggest improvements. Here is my code:
--read_int function
read_int :: IO Integer
read_int = do
line <- getLine
readIO line
--is_prime function
is_prime :: Integer -> Bool
is_prime n = helper n 2
where
helper :: Integer -> Integer -> Bool
helper n i
| n < 2 * i = True
| mod n i > 0 = helper n (i+1)
| otherwise = False
main = do
n <- read_int
dump 1 n
where
dump i x = do
putStrLn ( show (i) ++ " is a prime? " ++ show (is_prime i) )
if i >= x
then putStrLn ("")
else do
dump (i+1) x
You are misreading the tutorial. It would say the type signature should be
is_prime :: (Integral a) => a -> Bool
-- NOT Integer a
These are different types:
Integer -> Bool
This is a function that takes a value of type Integer and gives back a value of type Bool.
Integral a => a -> Bool
This is a function that takes a value of type a and gives back a value of type Bool.
What is a? It can be any type of the caller's choice that implements the Integral type class, such as Integer or Int.
(And the difference between Int and Integer? The latter can represent an integer of any magnitude, the former wraps around eventually, similar to ints in C/Java/etc.)
The idiomatic way to loop depends on what your loop does: it will either be a map, a fold, or a filter.
Your loop in main is a map, and because you're doing i/o in your loop, you need to use mapM_.
let dump i = putStrLn ( show (i) ++ " is a prime? " ++ show (is_prime i) )
in mapM_ dump [1..n]
Meanwhile, your loop in is_prime is a fold (specifically all in this case):
is_prime :: Integer -> Bool
is_prime n = all nondivisor [2 .. n `div` 2]
where
nondivisor :: Integer -> Bool
nondivisor i = mod n i > 0
(And on a minor point of style, it's conventional in Haskell to use names like isPrime instead of names like is_prime.)
Part 1: If you look at the tutorial again, you'll notice that it actually gives type signatures in the following forms:
isPrime :: Integer -> Bool
-- or
isPrime :: Integral a => a -> Bool
isPrime :: (Integral a) => a -> Bool -- equivalent
Here, Integer is the name of a concrete type (has an actual representation) and Integral is the name of a class of types. The Integer type is a member of the Integral class.
The constraint Integral a means that whatever type a happens to be, a has to be a member of the Integral class.
Part 2: There are plenty of ways to write such a function. Your recursive definition looks fine (although you might want to use n < i * i instead of n < 2 * i, since it's faster).
If you're learning Haskell, you'll probably want to try writing it using higher-order functions or list comprehensions. Something like:
module Main (main) where
import Control.Monad (forM_)
isPrime :: Integer -> Bool
isPrime n = all (\i -> (n `rem` i) /= 0) $ takeWhile (\i -> i^2 <= n) [2..]
main :: IO ()
main = do n <- readLn
forM_ [1..n] $ \i ->
putStrLn (show (i) ++ " is a prime? " ++ show (isPrime i))
It is Integral a, not Integer a. See http://www.haskell.org/haskellwiki/Converting_numbers.
map and friends is how you loop in Haskell. This is how I would re-write the loop:
main :: IO ()
main = do
n <- read_int
mapM_ tell_prime [1..n]
where tell_prime i = putStrLn (show i ++ " is a prime? " ++ show (is_prime i))
First day learning haskell, and coming from a python background I'm really having trouble debugging when it comes to type; Currently I'm working on a simple function to see if a number is a prime;
prime p = if p == 1 then False else if p == 2 then True else if maximum ([if p `mod` x == 0 then x else -1 | x<-[2..(floor(p**0.5))]]) > 0 then False else True
It works when I have a specific number instead of the generic P, but no matter what I try (and I've tried a lot, including just moving onto different problems) I always get some kind of error regarding type. For this current iteration, I'm getting the error
<interactive>:149:1: error:
* Ambiguous type variable `a0' arising from a use of `prime'
prevents the constraint `(RealFrac a0)' from being solved.
Probable fix: use a type annotation to specify what `a0' should be.
These potential instances exist:
instance RealFrac Double -- Defined in `GHC.Float'
instance RealFrac Float -- Defined in `GHC.Float'
...plus one instance involving out-of-scope types
(use -fprint-potential-instances to see them all)
* In the expression: prime 2
In an equation for `it': it = prime 2
<interactive>:149:7: error:
* Ambiguous type variable `a0' arising from the literal `2'
prevents the constraint `(Num a0)' from being solved.
Probable fix: use a type annotation to specify what `a0' should be.
These potential instances exist:
instance Num Integer -- Defined in `GHC.Num'
instance Num Double -- Defined in `GHC.Float'
instance Num Float -- Defined in `GHC.Float'
...plus two others
...plus one instance involving out-of-scope types
(use -fprint-potential-instances to see them all)
* In the first argument of `prime', namely `2'
In the expression: prime 2
In an equation for `it': it = prime 2
If someone could, as well as debugging this particular program, give me a heads up on how to think of haskell types, I'd be incredibly grateful. I've tried looking at learnyouahaskell but so far I've had no luck applying that.
In short: by using mod, floor, and (**) all at the same time, you restrict the type of p a lot, and Haskell fails to find a numerical type to call prime.
The main problem here is in the iterable of your list comprehension:
[2..(floor(p**0.5))]
Here you call p ** 0.5, but since (**) has type (**) :: Floating a => a -> a -> a, that thus means that p has to be an instance of a type that is an instance of the Floating typeclass, for example a Float. I guess you do not want that.
Your floor :: (RealFrac a, Integral b) => a -> b even makes it worse, since now p also has to be of a type that is an instance of the RealFrac typeclass.
On the other hand, you use mod :: Integral a => a -> a -> a, so it means that your p has to be Floating, as well as Integral, which are rather two disjunctive sets: although strictly speaking, we can define such a type, it is rather weird for a number to be both Integral and Floating at the same type. Float is for instance a Floating number, but not Integral, and Int is Integral, but not a Floating type.
We have to find a way to relax the constraints put on p. Since usually non-Integral numbers are no primes at all, we better thus aim to throw out floor and (**). The optimization to iterate up to the square root of p is however a good idea, but we will need to find other means to enforce that.
One way to do this is by using a takeWhile :: (a -> Bool) -> [a] -> [a] where we take elements, until the square of the numbers is greater than p, so we can rewrite the [2..(floor(p**0.5))] to:
takeWhile (\x -> x * x <= p) [2..]
We even can work only with odd elements and 2, by writing it as:
takeWhile (\x -> x * x <= p) (2:[3, 5..])
If we test this with a p that is for instance set to 99, we get:
Prelude> takeWhile (\x -> x * x <= 99) (2:[3, 5..])
[2,3,5,7,9]
If we plug that in, we relaxed the type:
prime p = if p == 1 then False else if p == 2 then True else if maximum ([if p `mod` x == 0 then x else -1 | x <- takeWhile (\x -> x * x <= p) (2:[3, 5..])]) > 0 then False else True
we actually relaxed it enough:
Prelude> :t prime
prime :: Integral a => a -> Bool
and we get:
Prelude> prime 23
True
But the code is very ugly and rather un-Haskell. First of all, you here use maximum as a trick to check if all elements satisfy a predicate. But it makes no sense to do that this way: from the moment one of the elements is dividable, we know that the number is not prime. So we can better use the all :: (a -> Bool) -> [a] -> Bool function. Furthermore conditions are usually checked by using pattern matching and guards, so we can write it like:
prime :: Integral a => a -> Bool
prime n | n < 2 = False
| otherwise = all ((/=) 0 . mod n) divisors
where divisors = takeWhile (\x -> x * x <= n) (2:[3, 5..])
Your code can be simplified as
prime p = if p == 1 then False else
if p == 2 then True else
if maximum ([if p `mod` x == 0 then x else -1 | x<-[2..(floor(p**0.5))]]) > 0
then False else True
=
prime p = if p == 1 then False else
if p == 2 then True else
not (maximum [if p `mod` x == 0 then x else -1 | x<-[2..floor(p**0.5)]] > 0 )
=
prime p = not ( p == 1 ) &&
( p == 2 ||
maximum [if p `mod` x == 0 then x else -1 | x<-[2..floor(p**0.5)]] <= 0 )
=
prime p = p /= 1 &&
( p == 2 ||
maximum [if p `mod` x == 0 then x else -1 | x<-[2..floor(p**0.5)]] == -1 )
=~
prime p = p == 2 || p > 2 && null [x | x <- [2..floor(p**0.5)], p `mod` x == 0]
(convince yourself in the validity of each transformation).
This still gives us a type error of course, because (**) :: Floating a => a -> a -> a and mod :: Integral a => a -> a -> a are conflicting. To counter that, just throw a fromIntegral in there:
isPrime :: Integral a => a -> Bool
isPrime p = p == 2 ||
p > 2 && null [x | x <- [2..floor(fromIntegral p**0.5)], p `mod` x == 0]
and it's working:
~> filter isPrime [1..100]
[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]
import Data.Function (on)
import Data.List (sort)
data Monomial = Monomial
{ m_coeff :: Coefficient
, m_powers :: [(Variable, Power)]
}
deriving ()
instance Ord Monomial where
(>=) = on (>=) m_powers
instance Eq Monomial where
(==) = on (==) m_powers
That's an excerpt from my code, cut down to principal size. Let's try comparing:
*Main> (Monomial 1 [("x",2)]) > (Monomial (-1) [])
/* Computation hangs here */
*Main> (Monomial 1 [("x",2)]) < (Monomial (-1) [])
/* Computation hangs here */
On a side note, it's interesting that if I replace s/(>=)/(>)/g in instance declaration, it will not hang on the fist pair, but still will on the second:
*Main> (Monomial 1 [("x",2)]) > (Monomial (-1) [])
True
*Main> (Monomial 1 [("x",2)]) < (Monomial (-1) [])
/* Computation hangs here */
Although the standard states minimal declaration of Eq instance to be either$compare$ or $(>=)$.
What might be the problem here? (>=) on lists seems to work just fine.
Short answer:
You need to provide either (<=) or compare to have a complete definition for Ord, not (>=).
Longer explanation:
It is common for type classes in Haskell to have default implementations of some methods implemented in terms of other methods. You can then choose which ones you want to implement. For example, Eq looks like this:
class Eq a where
(==), (/=) :: a -> a -> Bool
x /= y = not (x == y)
x == y = not (x /= y)
Here, you must either implement (==) or (/=), otherwise trying to use either of them will cause an infinite loop. Which methods you need to provide is usually listed as the minimal complete definition in the documentation.
The minimal complete definition for Ord instances, as listed in the documentation, is either (<=) or compare. Since you've only provided (>=), you have not provided a complete definition, and therefore some of the methods will loop. You can fix it by e.g. changing your instance to provide compare instead.
instance Ord Monomial where
compare = compare `on` m_powers
Let's look at the default instance for Ord:
class (Eq a) => Ord a where
compare :: a -> a -> Ordering
(<), (<=), (>), (>=) :: a -> a -> Bool
max, min :: a -> a -> a
compare x y = if x == y then EQ
-- NB: must be '<=' not '<' to validate the
-- above claim about the minimal things that
-- can be defined for an instance of Ord:
else if x <= y then LT
else GT
x < y = case compare x y of { LT -> True; _ -> False }
x <= y = case compare x y of { GT -> False; _ -> True }
x > y = case compare x y of { GT -> True; _ -> False }
x >= y = case compare x y of { LT -> False; _ -> True }
-- These two default methods use '<=' rather than 'compare'
-- because the latter is often more expensive
max x y = if x <= y then y else x
min x y = if x <= y then x else y
So, if you supply >= and == as above, only, then you are in trouble, since:
> is defined in terms of compare
But
compare is defined in terms of <=
<= is defined in terms of compare
So you have an infinite loop!
A minimum definition must defined <= or compare, not '>=`.
So I'm writing a program which returns a procedure for some given arithmetic problem, so I wanted to instance a couple of functions to Show so that I can print the same expression I evaluate when I test. The trouble is that the given code matches (-) to the first line when it should fall to the second.
{-# OPTIONS_GHC -XFlexibleInstances #-}
instance Show (t -> t-> t) where
show (+) = "plus"
show (-) = "minus"
main = print [(+),(-)]
returns
[plus,plus]
Am I just committing a mortal sin printing functions in the first place or is there some way I can get it to match properly?
edit:I realise I am getting the following warning:
Warning: Pattern match(es) are overlapped
In the definition of `show': show - = ...
I still don't know why it overlaps, or how to stop it.
As sepp2k and MtnViewMark said, you can't pattern match on the value of identifiers, only on constructors and, in some cases, implicit equality checks. So, your instance is binding any argument to the identifier, in the process shadowing the external definition of (+). Unfortunately, this means that what you're trying to do won't and can't ever work.
A typical solution to what you want to accomplish is to define an "arithmetic expression" algebraic data type, with an appropriate show instance. Note that you can make your expression type itself an instance of Num, with numeric literals wrapped in a "Literal" constructor, and operations like (+) returning their arguments combined with a constructor for the operation. Here's a quick, incomplete example:
data Expression a = Literal a
| Sum (Expression a) (Expression a)
| Product (Expression a) (Expression a)
deriving (Eq, Ord, Show)
instance (Num a) => Num (Expression a) where
x + y = Sum x y
x * y = Product x y
fromInteger x = Literal (fromInteger x)
evaluate (Literal x) = x
evaluate (Sum x y) = evaluate x + evaluate y
evaluate (Product x y) = evaluate x * evaluate y
integer :: Integer
integer = (1 + 2) * 3 + 4
expr :: Expression Integer
expr = (1 + 2) * 3 + 4
Trying it out in GHCi:
> integer
13
> evaluate expr
13
> expr
Sum (Product (Sum (Literal 1) (Literal 2)) (Literal 3)) (Literal 4)
Here's a way to think about this. Consider:
answer = 42
magic = 3
specialName :: Int -> String
specialName answer = "the answer to the ultimate question"
specialName magic = "the magic number"
specialName x = "just plain ol' " ++ show x
Can you see why this won't work? answer in the pattern match is a variable, distinct from answer at the outer scope. So instead, you'd have to write this like:
answer = 42
magic = 3
specialName :: Int -> String
specialName x | x == answer = "the answer to the ultimate question"
specialName x | x == magic = "the magic number"
specialName x = "just plain ol' " ++ show x
In fact, this is just what is going on when you write constants in a pattern. That is:
digitName :: Bool -> String
digitName 0 = "zero"
digitName 1 = "one"
digitName _ = "math is hard"
gets converted by the compiler to something equivalent to:
digitName :: Bool -> String
digitName x | x == 0 = "zero"
digitName x | x == 1 = "one"
digitName _ = "math is hard"
Since you want to match against the function bound to (+) rather than just bind anything to the symbol (+), you'd need to write your code as:
instance Show (t -> t-> t) where
show f | f == (+) = "plus"
show f | f == (-) = "minus"
But, this would require that functions were comparable for equality. And that is an undecidable problem in general.
You might counter that you are just asking the run-time system to compare function pointers, but at the language level, the Haskell programmer doesn't have access to pointers. In other words, you can't manipulate references to values in Haskell(*), only values themselves. This is the purity of Haskell, and gains referential transparency.
(*) MVars and other such objects in the IO monad are another matter, but their existence doesn't invalidate the point.
It overlaps because it treats (+) simply as a variable, meaning on the RHS the identifier + will be bound to the function you called show on.
There is no way to pattern match on functions the way you want.
Solved it myself with a mega hack.
instance (Num t) => Show (t -> t-> t) where
show op =
case (op 6 2) of
8 -> "plus"
4 -> "minus"
12 -> "times"
3 -> "divided"