Data.Ranged.RangedSet

Synopsis

## Ranged Set Type

data DiscreteOrdered v => RSet v Source

An RSet (for Ranged Set) is a list of ranges. The ranges must be sorted and not overlap.

Instances

 DiscreteOrdered v => Eq (RSet v) (Show v, DiscreteOrdered v) => Show (RSet v) (Arbitrary v, DiscreteOrdered v, Show v) => Arbitrary (RSet v) DiscreteOrdered a => Monoid (RSet a)

## Ranged Set construction functions and their preconditions

makeRangedSet :: DiscreteOrdered v => [Range v] -> RSet vSource

Create a new Ranged Set from a list of ranges. The list may contain ranges that overlap or are not in ascending order.

unsafeRangedSet :: DiscreteOrdered v => [Range v] -> RSet vSource

Create a new Ranged Set from a list of ranges. `validRangeList ranges` must return `True`. This precondition is not checked.

validRangeList :: DiscreteOrdered v => [Range v] -> BoolSource

Determine if the ranges in the list are both in order and non-overlapping. If so then they are suitable input for the unsafeRangedSet function.

normaliseRangeList :: DiscreteOrdered v => [Range v] -> [Range v]Source

Rearrange and merge the ranges in the list so that they are in order and non-overlapping.

rSingleton :: DiscreteOrdered v => v -> RSet vSource

Create a Ranged Set from a single element.

Arguments

 :: DiscreteOrdered a => Boundary a A first lower boundary. -> (Boundary a -> Boundary a) A function from a lower boundary to an upper boundary, which must return a result greater than the argument (not checked). -> (Boundary a -> Maybe (Boundary a)) A function from a lower boundary to `Maybe` the successor lower boundary, which must return a result greater than the argument (not checked). If ranges overlap then they will be merged. -> RSet a

Construct a range set.

## Predicates

rSetIsEmpty :: DiscreteOrdered v => RSet v -> BoolSource

True if the set has no members.

rSetIsFull :: DiscreteOrdered v => RSet v -> BoolSource

True if the negation of the set has no members.

(-?-) :: DiscreteOrdered v => RSet v -> v -> BoolSource

True if the value is within the ranged set. Infix precedence is left 5.

(-<=-) :: DiscreteOrdered v => RSet v -> RSet v -> BoolSource

True if the first argument is a subset of the second argument, or is equal.

Infix precedence is left 5.

(-<-) :: DiscreteOrdered v => RSet v -> RSet v -> BoolSource

True if the first argument is a strict subset of the second argument.

Infix precedence is left 5.

## Set Operations

(-\/-) :: DiscreteOrdered v => RSet v -> RSet v -> RSet vSource

Set union for ranged sets. Infix precedence is left 6.

(-/\-) :: DiscreteOrdered v => RSet v -> RSet v -> RSet vSource

Set intersection for ranged sets. Infix precedence is left 7.

(-!-) :: DiscreteOrdered v => RSet v -> RSet v -> RSet vSource

Set difference. Infix precedence is left 6.

rSetNegation :: DiscreteOrdered a => RSet a -> RSet aSource

Set negation.

## Useful Sets

The empty set.

The set that contains everything.

## QuickCheck Properties

### Construction

prop_validNormalised :: DiscreteOrdered a => [Range a] -> BoolSource

A normalised range list is valid for unsafeRangedSet

``` prop_validNormalised ls = validRangeList \$ normaliseRangeList ls
```

prop_has :: DiscreteOrdered a => [Range a] -> a -> BoolSource

Iff a value is in a range list then it is in a ranged set constructed from that list.

``` prop_has ls v = (ls `rangeListHas` v) == makeRangedSet ls -?- v
```

Verifies the correct membership of a set containing all integers starting with the digit "1" up to 19999.

``` prop_unfold = (v <= 99999 && head (show v) == '1') == (initial1 -?- v)
where
initial1 = rSetUnfold (BoundaryBelow 1) addNines times10
addNines (BoundaryBelow n) = BoundaryAbove \$ n * 2 - 1
times10 (BoundaryBelow n) =
if n <= 1000 then Just \$ BoundaryBelow \$ n * 10 else Nothing
```

### Basic Operations

prop_union :: DiscreteOrdered a => RSet a -> RSet a -> a -> BoolSource

Iff a value is in either of two ranged sets then it is in the union of those two sets.

``` prop_union rs1 rs2 v =
(rs1 -?- v || rs2 -?- v) == ((rs1 -\/- rs2) -?- v)
```

prop_intersection :: DiscreteOrdered a => RSet a -> RSet a -> a -> BoolSource

Iff a value is in both of two ranged sets then it is n the intersection of those two sets.

``` prop_intersection rs1 rs2 v =
(rs1 -?- v && rs2 -?- v) == ((rs1 -/\- rs2) -?- v)
```

prop_difference :: DiscreteOrdered a => RSet a -> RSet a -> a -> BoolSource

Iff a value is in ranged set 1 and not in ranged set 2 then it is in the difference of the two.

``` prop_difference rs1 rs2 v =
(rs1 -?- v && not (rs2 -?- v)) == ((rs1 -!- rs2) -?- v)
```

prop_negation :: DiscreteOrdered a => RSet a -> a -> BoolSource

Iff a value is not in a ranged set then it is in its negation.

``` prop_negation rs v = rs -?- v == not (rSetNegation rs -?- v)
```

prop_not_empty :: DiscreteOrdered a => RSet a -> a -> PropertySource

A set that contains a value is not empty

``` prop_not_empty rs v = (rs -?- v) ==> not (rSetIsEmpty rs)
```

### Some Identities and Inequalities

prop_empty :: DiscreteOrdered a => a -> BoolSource

The empty set has no members.

``` prop_empty v = not (rSetEmpty -?- v)
```

prop_full :: DiscreteOrdered a => a -> BoolSource

The full set has every member.

``` prop_full v = rSetFull -?- v
```

The intersection of a set with its negation is empty.

``` prop_empty_intersection rs =
rSetIsEmpty (rs -/\- rSetNegation rs)
```

The union of a set with its negation is full.

``` prop_full_union rs v =
rSetIsFull (rs -\/- rSetNegation rs)
```

prop_union_superset :: DiscreteOrdered a => RSet a -> RSet a -> BoolSource

The union of two sets is the non-strict superset of both.

``` prop_union_superset rs1 rs2 =
rs1 -<=- u && rs2 -<=- u
where
u = rs1 -\/- rs2
```

The intersection of two sets is the non-strict subset of both.

``` prop_intersection_subset rs1 rs2 =
i -<=- rs1 && i -<=- rs2
where
i = rs1 -/\- rs2
```

prop_diff_intersect :: DiscreteOrdered a => RSet a -> RSet a -> BoolSource

The difference of two sets intersected with the subtractand is empty.

``` prop_diff_intersect rs1 rs2 =
rSetIsEmpty ((rs1 -!- rs2) -/\- rs2)
```

prop_subset :: DiscreteOrdered a => RSet a -> BoolSource

A set is the non-strict subset of itself.

``` prop_subset rs = rs -<=- rs
```

A set is not the strict subset of itself.

``` prop_strict_subset rs = not (rs -<- rs)
```

If rs1 - rs2 is not empty then the union of rs1 and rs2 will be a strict superset of rs2.

``` prop_union_strict_superset rs1 rs2 =
(not \$ rSetIsEmpty (rs1 -!- rs2))
==> (rs2 -<- (rs1 -\/- rs2))
```

Intersection commutes.

``` prop_intersection_commutes rs1 rs2 = (rs1 -/\- rs2) == (rs2 -/\- rs1)
```

prop_union_commutes :: DiscreteOrdered a => RSet a -> RSet a -> BoolSource

Union commutes.

``` prop_union_commutes rs1 rs2 = (rs1 -\/- rs2) == (rs2 -\/- rs1)
```

prop_intersection_associates :: DiscreteOrdered a => RSet a -> RSet a -> RSet a -> BoolSource

Intersection associates.

``` prop_intersection_associates rs1 rs2 rs3 =
((rs1 -/\- rs2) -/\- rs3) == (rs1 -/\- (rs2 -/\- rs3))
```

prop_union_associates :: DiscreteOrdered a => RSet a -> RSet a -> RSet a -> BoolSource

Union associates.

``` prop_union_associates rs1 rs2 rs3 =
((rs1 -\/- rs2) -\/- rs3) == (rs1 -\/- (rs2 -\/- rs3))
```

De Morgan's Law for Intersection.

``` prop_de_morgan_intersection rs1 rs2 =
rSetNegation (rs1 -/\- rs2) == (rSetNegation rs1 -\/- rSetNegation rs2)
```

prop_de_morgan_union :: DiscreteOrdered a => RSet a -> RSet a -> BoolSource

De Morgan's Law for Union.

``` prop_de_morgan_union rs1 rs2 =
rSetNegation (rs1 -\/- rs2) == (rSetNegation rs1 -/\- rSetNegation rs2)
```