MAYBE Problem: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) Proof: DP Processor: DPs: a__terms#(N) -> mark#(N) a__terms#(N) -> a__sqr#(mark(N)) a__add#(0(),X) -> mark#(X) a__first#(s(X),cons(Y,Z)) -> mark#(Y) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) mark#(sqr(X)) -> mark#(X) mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(dbl(X)) -> mark#(X) mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) TDG Processor: DPs: a__terms#(N) -> mark#(N) a__terms#(N) -> a__sqr#(mark(N)) a__add#(0(),X) -> mark#(X) a__first#(s(X),cons(Y,Z)) -> mark#(Y) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) mark#(sqr(X)) -> mark#(X) mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(dbl(X)) -> mark#(X) mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) graph: a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(recip(X)) -> mark#(X) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(first(X1,X2)) -> mark#(X1) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(first(X1,X2)) -> mark#(X2) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(dbl(X)) -> a__dbl#(mark(X)) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(dbl(X)) -> mark#(X) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(add(X1,X2)) -> mark#(X1) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(add(X1,X2)) -> mark#(X2) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(sqr(X)) -> a__sqr#(mark(X)) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(sqr(X)) -> mark#(X) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(terms(X)) -> a__terms#(mark(X)) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(terms(X)) -> mark#(X) a__add#(0(),X) -> mark#(X) -> mark#(recip(X)) -> mark#(X) a__add#(0(),X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__add#(0(),X) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) a__add#(0(),X) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) a__add#(0(),X) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) a__add#(0(),X) -> mark#(X) -> mark#(dbl(X)) -> a__dbl#(mark(X)) a__add#(0(),X) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) a__add#(0(),X) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) a__add#(0(),X) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) a__add#(0(),X) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) a__add#(0(),X) -> mark#(X) -> mark#(sqr(X)) -> a__sqr#(mark(X)) a__add#(0(),X) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) a__add#(0(),X) -> mark#(X) -> mark#(terms(X)) -> a__terms#(mark(X)) a__add#(0(),X) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) -> a__first#(s(X),cons(Y,Z)) -> mark#(Y) mark#(first(X1,X2)) -> mark#(X2) -> mark#(recip(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(first(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X2) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(first(X1,X2)) -> mark#(X2) -> mark#(dbl(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(first(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X2) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(first(X1,X2)) -> mark#(X2) -> mark#(sqr(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(first(X1,X2)) -> mark#(X2) -> mark#(terms(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(recip(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(first(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(first(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(first(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(first(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(first(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> mark#(X) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) -> a__add#(0(),X) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(recip(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X2) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(dbl(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X2) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(terms(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(recip(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(dbl(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(dbl(X)) -> mark#(X) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(dbl(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(dbl(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(dbl(X)) -> mark#(X) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(dbl(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(dbl(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(sqr(X)) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(sqr(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(sqr(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(sqr(X)) -> mark#(X) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(sqr(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(sqr(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(sqr(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(sqr(X)) -> mark#(X) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(sqr(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(sqr(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(recip(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(terms(X)) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(terms(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(terms(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(terms(X)) -> mark#(X) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(terms(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(terms(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(terms(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(terms(X)) -> mark#(X) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(terms(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(terms(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) -> a__terms#(N) -> a__sqr#(mark(N)) mark#(terms(X)) -> a__terms#(mark(X)) -> a__terms#(N) -> mark#(N) mark#(recip(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(recip(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(recip(X)) -> mark#(X) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(recip(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(recip(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(recip(X)) -> mark#(X) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(recip(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(recip(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) a__terms#(N) -> mark#(N) -> mark#(recip(X)) -> mark#(X) a__terms#(N) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1) a__terms#(N) -> mark#(N) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) a__terms#(N) -> mark#(N) -> mark#(first(X1,X2)) -> mark#(X1) a__terms#(N) -> mark#(N) -> mark#(first(X1,X2)) -> mark#(X2) a__terms#(N) -> mark#(N) -> mark#(dbl(X)) -> a__dbl#(mark(X)) a__terms#(N) -> mark#(N) -> mark#(dbl(X)) -> mark#(X) a__terms#(N) -> mark#(N) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) a__terms#(N) -> mark#(N) -> mark#(add(X1,X2)) -> mark#(X1) a__terms#(N) -> mark#(N) -> mark#(add(X1,X2)) -> mark#(X2) a__terms#(N) -> mark#(N) -> mark#(sqr(X)) -> a__sqr#(mark(X)) a__terms#(N) -> mark#(N) -> mark#(sqr(X)) -> mark#(X) a__terms#(N) -> mark#(N) -> mark#(terms(X)) -> a__terms#(mark(X)) a__terms#(N) -> mark#(N) -> mark#(terms(X)) -> mark#(X) Restore Modifier: DPs: a__terms#(N) -> mark#(N) a__terms#(N) -> a__sqr#(mark(N)) a__add#(0(),X) -> mark#(X) a__first#(s(X),cons(Y,Z)) -> mark#(Y) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) mark#(sqr(X)) -> mark#(X) mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(dbl(X)) -> mark#(X) mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) SCC Processor: #sccs: 1 #rules: 15 #arcs: 172/324 DPs: a__first#(s(X),cons(Y,Z)) -> mark#(Y) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) a__terms#(N) -> mark#(N) mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) a__add#(0(),X) -> mark#(X) mark#(dbl(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) Open