Multi-line Function Definitions

Local Definitions

For example:

res = f * (f - y)  where
 
    y = 1 / 4
 
    f = 6 * y

Syntax:

Semantics: Limited scope for the variables (y, f).

Local definitions are definitions of constants or functions with limited scope.

Syntax of local definitions:

The res is a global definition, that is, it is visible through the entire module. f and y are local definitions, they can be only visible for each other and the definition before where.

It is recommended to place the where at end of the line. It may be also written to the following line, but it must be indented one level higher than the preceding definition in that case.

Local definitions are recommended to use when both of the following criteria are satisfied at the same time.

Example

Definition of function composition by introducing local definitions.

(.) :: (b -> c) -> (a -> b) -> (a -> c)

f . g = h  where
  h x = f (g x)

Definition of function composition without local definitions.

(.) :: (b -> c) -> (a -> b) -> (a -> c)
(f . g) x = f (g x)

Constant Patterns

When defining constans, arbitrary patterns may be written in place of the name.

[one,two,three,four,five] = [1..5]

There is now five constants are defined this way.

Test>
1 :: Integer
Test>
2 :: Integer
Test>
3 :: Integer
Test>
4 :: Integer
Test>
5 :: Integer

Application

Constants patterns can be very beneficial, especially when pattern matching on results of functions returning n-tuples is required.

For example:

unzip :: [(a, b)] -> ([a], [b])

Test>
([1, 3, 4], "xzv") :: ([Integer], [Char])

Exercise: Prelude.splitAt [*]

Redefine the splitAt function. Give a more efficient version of the definition below.

-- splitAt n l = (take n l, drop n l)
splitAt :: Int -> [a] -> ([a], [a])

Test>
([1, 2, 3], [4, 5, 6, 7, 8, 9]) :: ([Integer], [Integer])
Test>
([], [1, 2, 3, 4, 5, 6, 7, 8, 9]) :: ([Integer], [Integer])
Test>
([1, 2, 3, 4, 5, 6, 7, 8, 9], []) :: ([Integer], [Integer])

Exercise: Split List

split :: [a] -> ([a], [a])

Test>
([1, 3, 5, 7], [2, 4, 6, 8]) :: ([Integer], [Integer])
Test>
([1, 3, 5, 7, 9], [2, 4, 6, 8]) :: ([Integer], [Integer])

Exercise: Merge of Ordered Lists

ordMerge :: Ord a => [a] -> [a] -> [a]

Test>
[1, 2, 4, 4, 5, 6, 7, 8] :: [Integer]

Exercise: Sort Merge [*]

Define a sort merge. The algorithm is as follows. Split list with more than one element, then merge the two partitions after they have been sorted.

msort :: Ord a => [a] -> [a]
 -- split, ordMerge

Test>
[0, 1, 2, 3, 4, 6, 11] :: [Integer]
Test>
[1, 1, 2, 2] :: [Integer]

Exercise: Draw Triangle

Draw a triangle with sides of a, b, and c in the way given by the figure below.

(0,0)(b,0)(a*cos(γ),a*sin(γ))bac

Hint: cos(γ) = (a2 + b2 - c2) / (2ab).

It is recommended to calculate the value of cos(y) locally then derive the value of sin(y) from that.

triangleFromSides :: Double -> Double -> Double -> Diagram

Test>