/* Webber Assignment 8
   Dan J. Fraser (1219229) */
   
list_of_k_integers(X, List) :- 
   counter(X,0,List).

counter(Size, Count, List) :-
 list_of_k_integers_that_sum_to_S(Size,Count,List).

counter(Size,Count,List) :-
 plus(Count,1,New_Count),
 counter(Size,New_Count,List).

an_integer_less_than_or_equal_to(X, X) :- .

an_integer_less_than_or_equal_to(Int, X) :- 
  plus(Z, 1, Int), an_integer_less_than_or_equal_to(Z,X).

list_of_k_integers_that_sum_to_S(0, 0, nil) :- .

list_of_k_integers_that_sum_to_S(Size, 0, 0:Tail) :- 
  not Size = 0,
  plus(New_Size,1,Size),
  list_of_k_integers_that_sum_to_S(New_Size, 0, Tail).

list_of_k_integers_that_sum_to_S(Size, Sum, Y:X ) :- 
  not Size = 0,
  not Sum = 0,
  an_integer_less_than_or_equal_to(Sum,Y),
  plus(New_Size,1,Size),
  plus(New_Sum,Y,Sum),
  list_of_k_integers_that_sum_to_S(New_Size, New_Sum, X).