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For all set types `S`

that
are models concept `Set`

(`itl::set`

, `interval_set`

,
`separate_interval_set`

and `split_interval_set`

)
most of the well known mathematical laws
on sets were successfully checked via LaBatea. The next tables are
giving an overview over the checked laws ordered by operations. If possible,
the laws are formulated with the stronger lexicographical equality (`operator ==`

)
which implies the law's validity for the weaker element equality `is_element_equal`

. Throughout this chapter
we will denote element equality as `=e=`

instead
of `is_element_equal`

where
a short notation is advantageous.

For the operation * set union*
available as

```
operator +,
+=, |, |=
```

and the neutral element `neutron<S>::value()`

which is the empty set `S()`

these laws hold:
Associativity<S,+,== >: S a,b,c; a+(b+c) == (a+b)+c Neutrality<S,+,== > : S a; a+S() == a Commutativity<S,+,== >: S a,b; a+b == b+a

For the operation * set intersection*
available as

```
operator &,
&=
```

these laws were validated:

Associativity<S,&,== >: S a,b,c; a&(b&c) == (a&b)&c Commutativity<S,&,== >: S a,b; a&b == b&a

For set difference there are only these laws. It is not associative and not commutative. It's neutrality is non symmetrical.

RightNeutrality<S,-,== > : S a; a-S() == a Inversion<S,-,== >: S a; a - a == S()

Summarized in the next table are laws that use `+`

,
`&`

and `-`

as a single operation. For all validated laws, the left and right hand sides
of the equations are lexicographically equal, as denoted by `==`

in the cells of the table.

+ & - Associativity == == Neutrality == == Commutativity == == Inversion ==

Laws, like distributivity, that use more than one operation can sometimes
be instantiated for different sequences of operators as can be seen below.
In the two instantiations of the distributivity laws operators `+`

and `&`

are swapped. So we can have small operator signatures like `+,&`

and `&,+`

to describe such instantiations, which will be used below. Not all instances
of distributivity laws hold for lexicographical equality. Therefore they
are denoted using a *variable* equality `=v=`

below.

Distributivity<S,+,&,=v= > : S a,b,c; a + (b & c) =v= (a + b) & (a + c) Distributivity<S,&,+,=v= > : S a,b,c; a & (b + c) =v= (a & b) + (a & c) RightDistributivity<S,+,-,=v= > : S a,b,c; (a + b) - c =v= (a - c) + (b - c) RightDistributivity<S,&,-,=v= > : S a,b,c; (a & b) - c =v= (a - c) & (b - c)

The next table shows the relationship between law instances, interval combining style and the used equality relation.

+,& &,+ Distributivity joining == == separating == == splitting =e= =e= +,- &,- RightDistributivity joining == == separating == == splitting =e= ==

The table gives an overview over 12 instantiations of the four distributivity
laws and shows the equalities which the instantiations holds for. For instance
`RightDistributivity`

with
operator signature `+,-`

instantiated
for `split_interval_sets`

holds only for element equality (denoted as `=e=`

):

RightDistributivity<S,+,-,=e= > : S a,b,c; (a + b) - c =e= (a - c) + (b - c)

The remaining five instantiations of `RightDistributivity`

are valid for lexicographical equality (demoted as `==`

)
as well.

Interval combining styles correspond to containers according to

style set joining interval_set, itl::set separating separate_interval_set splitting split_interval_set

where the `itl::set`

of elements
can be subsumed in the `joining`

rows.

Finally there are two laws that combine all three major set operations: De Mogans Law and Symmetric Difference.

De Morgans Law is better known in an incarnation where the unary complement
operation `~`

is used. ```
~(a+b) ==
~a * ~b
```

.
The version below is an adaption for the binary set difference `-`

, which is also called * relative complement*.

DeMorgan<S,+,&,=v= > : S a,b,c; a - (b + c) =v= (a - b) & (a - c) DeMorgan<S,&,+,=v= > : S a,b,c; a - (b & c) =v= (a - b) + (a - c)

+,& &,+ DeMorgan joining == == separating == =e= splitting == =e=

Again not all law instances are valid for lexicographical equality. The second instantiations only holds for element equality, if the interval sets are non joining.

SymmetricDifference<S,== > : S a,b,c; (a + b) - (a & b) == (a - b) + (b - a)

Finally Symmetric Difference holds for all of itl set types and lexicographical equality.