YES 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) 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) Matrix Interpretation Processor: dim=1 interpretation: [a__first#](x0, x1) = x0 + 2x1, [a__add#](x0, x1) = x0 + x1, [mark#](x0) = x0, [a__terms#](x0) = x0 + 4, [first](x0, x1) = 2x0 + 7x1 + 1, [nil] = 4, [a__first](x0, x1) = 2x0 + 7x1 + 2, [a__add](x0, x1) = 2x0 + 5x1 + 1, [a__dbl](x0) = x0 + 4, [add](x0, x1) = 2x0 + 5x1 + 1, [dbl](x0) = x0 + 4, [sqr](x0) = x0 + 2, [0] = 5, [cons](x0, x1) = x0 + 1, [terms](x0) = 2x0 + 5, [s](x0) = 0, [recip](x0) = x0 + 1, [a__sqr](x0) = x0 + 3, [mark](x0) = 2x0, [a__terms](x0) = 2x0 + 5 orientation: a__first#(s(X),cons(Y,Z)) = 2Y + 2 >= Y = mark#(Y) mark#(terms(X)) = 2X + 5 >= X = mark#(X) mark#(terms(X)) = 2X + 5 >= 2X + 4 = a__terms#(mark(X)) a__terms#(N) = N + 4 >= N = mark#(N) mark#(sqr(X)) = X + 2 >= X = mark#(X) mark#(add(X1,X2)) = 2X1 + 5X2 + 1 >= X2 = mark#(X2) mark#(add(X1,X2)) = 2X1 + 5X2 + 1 >= X1 = mark#(X1) mark#(add(X1,X2)) = 2X1 + 5X2 + 1 >= 2X1 + 2X2 = a__add#(mark(X1),mark(X2)) a__add#(0(),X) = X + 5 >= X = mark#(X) mark#(dbl(X)) = X + 4 >= X = mark#(X) mark#(first(X1,X2)) = 2X1 + 7X2 + 1 >= X2 = mark#(X2) mark#(first(X1,X2)) = 2X1 + 7X2 + 1 >= X1 = mark#(X1) mark#(first(X1,X2)) = 2X1 + 7X2 + 1 >= 2X1 + 4X2 = a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(recip(X)) = X + 1 >= X = mark#(X) a__terms(N) = 2N + 5 >= 2N + 5 = cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) = 8 >= 5 = 0() a__sqr(s(X)) = 3 >= 0 = s(add(sqr(X),dbl(X))) a__dbl(0()) = 9 >= 5 = 0() a__dbl(s(X)) = 4 >= 0 = s(s(dbl(X))) a__add(0(),X) = 5X + 11 >= 2X = mark(X) a__add(s(X),Y) = 5Y + 1 >= 0 = s(add(X,Y)) a__first(0(),X) = 7X + 12 >= 4 = nil() a__first(s(X),cons(Y,Z)) = 7Y + 9 >= 2Y + 1 = cons(mark(Y),first(X,Z)) mark(terms(X)) = 4X + 10 >= 4X + 5 = a__terms(mark(X)) mark(sqr(X)) = 2X + 4 >= 2X + 3 = a__sqr(mark(X)) mark(add(X1,X2)) = 4X1 + 10X2 + 2 >= 4X1 + 10X2 + 1 = a__add(mark(X1),mark(X2)) mark(dbl(X)) = 2X + 8 >= 2X + 4 = a__dbl(mark(X)) mark(first(X1,X2)) = 4X1 + 14X2 + 2 >= 4X1 + 14X2 + 2 = a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) = 2X1 + 2 >= 2X1 + 1 = cons(mark(X1),X2) mark(recip(X)) = 2X + 2 >= 2X + 1 = recip(mark(X)) mark(s(X)) = 0 >= 0 = s(X) mark(0()) = 10 >= 5 = 0() mark(nil()) = 8 >= 4 = nil() a__terms(X) = 2X + 5 >= 2X + 5 = terms(X) a__sqr(X) = X + 3 >= X + 2 = sqr(X) a__add(X1,X2) = 2X1 + 5X2 + 1 >= 2X1 + 5X2 + 1 = add(X1,X2) a__dbl(X) = X + 4 >= X + 4 = dbl(X) a__first(X1,X2) = 2X1 + 7X2 + 2 >= 2X1 + 7X2 + 1 = first(X1,X2) problem: DPs: 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) Qed