:- initializing, add_function(eval, 1).
:- initializing, add_function(evalLazy, 1).
:- initializing, add_function(+, 2).
:- initializing, add_function(-, 2).
:- initializing, add_function(ones, 0).
:- initializing, add_function(nth_member, 2).
:- initializing, add_function(front, 2).
:- initializing, add_function(prime, 1).
:- initializing, add_function(primes, 1).
:- initializing, add_function(primes, 0).
:- initializing, add_function(ints, 1).
:- initializing, add_function(sift, 1).
:- initializing, add_function(filter, 2).
:- initializing, add_function(fib, 1).
:- initializing, add_function(fib, 0).
:- initializing, add_function(add_lists, 2).
:- initializing, add_function(concat, 2).
:- initializing, add_function(p, 2).
+(A9, B9, A) :-
	evalLazy(A9, B),
	evalLazy(B9, C),
	A is B + C.
-(A9, B9, A) :-
	evalLazy(A9, B),
	evalLazy(B9, C),
	A is B - C.

ones($lazy($lazy$ones(A), A)).

:- $lazy$ones(A) when ever.
$lazy$ones([1|A]) :-
	ones(A).
:- nth_member(A, [B|C], D) when ever.
:- nth_member(C, $lazy(B, A), D) when ever.
nth_member(C, $lazy(B, A), D):- call(B), nth_member(C, A, D).
nth_member(A9, [B9|C9], A) :-
	evalLazy(A9, B),
	(if B =:= 1 then
		A = B9
	else
		evalLazy(A9, C),
		evalLazy(1, D),
		E is C - D,
		nth_member(E, C9, F),
		A = F
	).
:- front(A, [], B) when ever.
:- front(C, $lazy(B, A), D) when ever.
front(C, $lazy(B, A), D):- call(B), front(C, A, D).
front(A9, [], []).
:- front(A, [B|C], D) when ever.
front(A9, [B9|C9], A) :-
	evalLazy(A9, B),
	(if B > 0 then
		evalLazy(A9, C),
		evalLazy(1, D),
		E is C - D,
		front(E, C9, F),
		A = [B9|F]
	else
		A = []
	).
:- prime(A, B) when ever.
prime(A9, A) :-
	primes(B),
	nth_member(A9, B, A).
:- primes(A, B) when ever.
primes(A9, A) :-
	primes(B),
	front(A9, B, A).

primes($lazy($lazy$primes(A), A)).

:- $lazy$primes(A) when ever.
$lazy$primes(A) :-
	ints(2, B),
	sift(B, A).

ints(A, $lazy($lazy$ints(A, B), B)).

:- $lazy$ints(A, B) when ever.
$lazy$ints(A9, [A9|A]) :-
	evalLazy(A9, B),
	evalLazy(1, C),
	D is B + C,
	ints(D, A).

sift(A, $lazy($lazy$sift(A, B), B)).

:- $lazy$sift([A|B], C) when ever.
:- $lazy$sift($lazy(B, A), C) when ever.
$lazy$sift($lazy(B, A), C):- call(B), $lazy$sift(A, C).
$lazy$sift([A9|B9], [A9|A]) :-
	filter(A9, B9, B),
	sift(B, A).

filter(A, B, $lazy($lazy$filter(A, B, C), C)).

:- $lazy$filter(A, [B|C], D) when ever.
:- $lazy$filter(C, $lazy(B, A), D) when ever.
$lazy$filter(C, $lazy(B, A), D):- call(B), $lazy$filter(C, A, D).
$lazy$filter(A9, [B9|C9], A) :-
	evalLazy(B9, B),
	evalLazy(A9, C),
	(if B mod C =:= 0 then
		filter(A9, C9, D),
		A = D
	else
		filter(A9, C9, E),
		A = [B9|E]
	).
:- fib(A, B) when ever.
fib(A9, A) :-
	fib(B),
	front(A9, B, A).
fib([0, 1|A]) :-
	add_lists([0, 1|A], [1|A], A).

add_lists(A, B, $lazy($lazy$add_lists(A, B, C), C)).

:- $lazy$add_lists([], [], A) when ever.
:- $lazy$add_lists($lazy(B, A), C, D) when ever.
$lazy$add_lists($lazy(B, A), C, D):- call(B), $lazy$add_lists(A, C, D).
:- $lazy$add_lists([], $lazy(B, A), C) when ever.
$lazy$add_lists([], $lazy(B, A), C):- call(B), $lazy$add_lists([], A, C).
$lazy$add_lists([], [], []).
:- $lazy$add_lists([A|B], [C|D], E) when ever.
:- $lazy$add_lists([C|D], $lazy(B, A), E) when ever.
$lazy$add_lists([C|D], $lazy(B, A), E):- call(B), $lazy$add_lists([C|D], A, E).
$lazy$add_lists([A9|B9], [C9|D9], [A|B]) :-
	evalLazy(A9, C),
	evalLazy(C9, D),
	A is C + D,
	add_lists(B9, D9, B).
:- concat([], A, B) when ever.
:- concat($lazy(B, A), C, D) when ever.
concat($lazy(B, A), C, D):- call(B), concat(A, C, D).
concat([], A9, A9).
:- concat([A|B], C, D) when ever.
concat([A9|B9], C9, [A9|A]) :-
	concat(B9, C9, A).
:- p(f(a), A, B) when ever.
:- p($lazy(B, A), C, D) when ever.
p($lazy(B, A), C, D):- call(B), p(A, C, D).
:- p(f($lazy(B, A)), C, D) when ever.
p(f($lazy(B, A)), C, D):- call(B), p(f(A), C, D).
p(f(a), A9, a).
:- p(b, a, A) when ever.
:- p(b, $lazy(B, A), C) when ever.
p(b, $lazy(B, A), C):- call(B), p(b, A, C).
p(b, a, b).