flatten([], []).
flatten([First|Rest], [First|RestFlattened]) :-
atomic(First),
flatten(Rest, RestFlattened).

flatten([First|Rest], FlattenedList) :-
flatten(First, FirstFlattened),
flatten(Rest, RestFlattened),
append(FirstFlattened, RestFlattened, FlattenedList).

insertionsort([],[]).
insertionsort([First|Rest], SortedList) :-
insertionsort(Rest,NewList),
insert(First, NewList, SortedList).

insert(Element, [], [Element]).
insert(Element, [First|Rest], [First|NewList]) :-
Element > First, insert(Element, Rest, NewList).
insert(Element, List, [Element|List]).

selectionsort([],[]).
selectionsort([First|Rest], [Smallest|SortedList]) :-
smallest(Rest, First, Smallest),
remove([First|Rest], Smallest, NewList),
selectionsort(NewList, SortedList).

/* looks for the smallest element in the list
atom A is the current smallest
*/
smallest([], Smallest,Smallest).
smallest([First|Rest], CurrSmallest, Smallest) :-
First < CurrSmallest, smallest(Rest, First, Smallest).
smallest([_|Rest], CurrSmallest, Smallest) :-
smallest(Rest, CurrSmallest, Smallest).

/* remove the first occurance of atom A from L */
remove([], _, []).
remove([First|Rest], First, Rest).
remove([First|Rest], Element, [First|NewList]) :-
remove(Rest, Element, NewList).

mergesort([],[]).
mergesort([OneElement], [OneElement]).
mergesort([OneElement,SecondElement], SortedList) :-
mergelists([OneElement],[SecondElement], SortedList).
mergesort([FirstLis|RestLis], SortedList) :-
split([FirstLis|RestLis],[FirstSplitLis,RestSplitLis]),
mergesort(FirstSplitLis, SortedFirstLis),
mergesort(RestSplitLis, SortedRestLis),
mergelists(SortedFirstLis, SortedRestLis, SortedList).

mergelists([], M, M).
mergelists(Lis, [], Lis).
/* assume L and M are sorted lists*/
mergelists([FirstLis|RestLis], [FirstM|RestM], [FirstLis|RestLisMerged])
:-
FirstLis < FirstM,
mergelists(RestLis, [FirstM|RestM], RestLisMerged).
mergelists([FirstLis|RestLis], [FirstM|RestM], [FirstM|RestMMerged]) :-
mergelists([FirstLis|RestLis], RestM, RestMMerged).

/* extract elements start to stop into a list */
sublist([],_,_,_,[]).
sublist([_|RestLis], Start, Stop, Ctr, SubList) :-
Ctr < Start,
NextCtr is Ctr + 1,
sublist(RestLis, Start, Stop, NextCtr, SubList).
sublist(_, _, Stop, Ctr, []) :-
Ctr > Stop.
sublist([FirstLis|RestLis], Start, Stop, Ctr, [FirstLis| SubList]) :-
NextCtr is Ctr + 1,
sublist(RestLis, Start, Stop, NextCtr, SubList).
/* split the list in half:, ; returns ((first half)(second half)) */
split([], [[],[]]).
split([OneElement],[[OneElement],[]]).
split(Lis, [SubLis1,SubLis2]) :-
length(Lis,Len),
HalfLen is Len//2,
HalfLenPlus is HalfLen + 1,
sublist(Lis,1,HalfLen, 1, SubLis1),
sublist(Lis, HalfLenPlus, Len, 1, SubLis2).

gcd(Num1, 0, Num1).
gcd(Num1, Num2, GCD) :-
Rem is Num1 mod Num2,
gcd(Num2, Rem, GCD).