YES Problem: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Proof: DP Processor: DPs: a__fst#(s(X),cons(Y,Z)) -> mark#(Y) a__from#(X) -> mark#(X) a__add#(0(),X) -> mark#(X) mark#(fst(X1,X2)) -> mark#(X2) mark#(fst(X1,X2)) -> mark#(X1) mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(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#(len(X)) -> mark#(X) mark#(len(X)) -> a__len#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) TDG Processor: DPs: a__fst#(s(X),cons(Y,Z)) -> mark#(Y) a__from#(X) -> mark#(X) a__add#(0(),X) -> mark#(X) mark#(fst(X1,X2)) -> mark#(X2) mark#(fst(X1,X2)) -> mark#(X1) mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(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#(len(X)) -> mark#(X) mark#(len(X)) -> a__len#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) graph: a__add#(0(),X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__add#(0(),X) -> mark#(X) -> mark#(len(X)) -> a__len#(mark(X)) a__add#(0(),X) -> mark#(X) -> mark#(len(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#(from(X)) -> a__from#(mark(X)) a__add#(0(),X) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__add#(0(),X) -> mark#(X) -> mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) a__add#(0(),X) -> mark#(X) -> mark#(fst(X1,X2)) -> mark#(X1) a__add#(0(),X) -> mark#(X) -> mark#(fst(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(len(X)) -> a__len#(mark(X)) a__from#(X) -> mark#(X) -> mark#(len(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) a__from#(X) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) a__from#(X) -> mark#(X) -> mark#(fst(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(fst(X1,X2)) -> mark#(X2) mark#(len(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(len(X)) -> mark#(X) -> mark#(len(X)) -> a__len#(mark(X)) mark#(len(X)) -> mark#(X) -> mark#(len(X)) -> mark#(X) mark#(len(X)) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(len(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(len(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(len(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(len(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(len(X)) -> mark#(X) -> mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) mark#(len(X)) -> mark#(X) -> mark#(fst(X1,X2)) -> mark#(X1) mark#(len(X)) -> mark#(X) -> mark#(fst(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) -> a__add#(0(),X) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X2) -> mark#(len(X)) -> a__len#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(len(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#(from(X)) -> a__from#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(fst(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X2) -> mark#(fst(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(len(X)) -> a__len#(mark(X)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(len(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#(from(X)) -> a__from#(mark(X)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(fst(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(fst(X1,X2)) -> mark#(X2) mark#(from(X)) -> a__from#(mark(X)) -> a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(len(X)) -> a__len#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(len(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) -> mark#(fst(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(fst(X1,X2)) -> mark#(X2) mark#(fst(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(fst(X1,X2)) -> mark#(X2) -> mark#(len(X)) -> a__len#(mark(X)) mark#(fst(X1,X2)) -> mark#(X2) -> mark#(len(X)) -> mark#(X) mark#(fst(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(fst(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X1) mark#(fst(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X2) mark#(fst(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(fst(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(fst(X1,X2)) -> mark#(X2) -> mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) mark#(fst(X1,X2)) -> mark#(X2) -> mark#(fst(X1,X2)) -> mark#(X1) mark#(fst(X1,X2)) -> mark#(X2) -> mark#(fst(X1,X2)) -> mark#(X2) mark#(fst(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(fst(X1,X2)) -> mark#(X1) -> mark#(len(X)) -> a__len#(mark(X)) mark#(fst(X1,X2)) -> mark#(X1) -> mark#(len(X)) -> mark#(X) mark#(fst(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(fst(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(fst(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X2) mark#(fst(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(fst(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(fst(X1,X2)) -> mark#(X1) -> mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) mark#(fst(X1,X2)) -> mark#(X1) -> mark#(fst(X1,X2)) -> mark#(X1) mark#(fst(X1,X2)) -> mark#(X1) -> mark#(fst(X1,X2)) -> mark#(X2) mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) -> a__fst#(s(X),cons(Y,Z)) -> mark#(Y) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(len(X)) -> a__len#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(len(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#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(fst(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(fst(X1,X2)) -> mark#(X2) a__fst#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__fst#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(len(X)) -> a__len#(mark(X)) a__fst#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(len(X)) -> mark#(X) a__fst#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) a__fst#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(add(X1,X2)) -> mark#(X1) a__fst#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(add(X1,X2)) -> mark#(X2) a__fst#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(from(X)) -> a__from#(mark(X)) a__fst#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(from(X)) -> mark#(X) a__fst#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) a__fst#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(fst(X1,X2)) -> mark#(X1) a__fst#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(fst(X1,X2)) -> mark#(X2) SCC Processor: #sccs: 1 #rules: 13 #arcs: 113/196 DPs: a__add#(0(),X) -> mark#(X) mark#(fst(X1,X2)) -> mark#(X2) mark#(fst(X1,X2)) -> mark#(X1) mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) a__fst#(s(X),cons(Y,Z)) -> mark#(Y) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(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)) mark#(len(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Arctic Interpretation Processor: dimension: 1 usable rules: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) interpretation: [a__add#](x0, x1) = x0 + x1, [a__from#](x0) = x0 + 0, [mark#](x0) = x0, [a__fst#](x0, x1) = 1x1, [len](x0) = x0 + 0, [a__len](x0) = x0 + 0, [add](x0, x1) = x0 + 2x1, [a__add](x0, x1) = x0 + 2x1, [from](x0) = x0 + 1, [a__from](x0) = x0 + 1, [fst](x0, x1) = 2x0 + 1x1, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [s](x0) = x0 + 0, [nil] = 0, [a__fst](x0, x1) = 2x0 + 1x1, [0] = 0 orientation: a__add#(0(),X) = X + 0 >= X = mark#(X) mark#(fst(X1,X2)) = 2X1 + 1X2 >= X2 = mark#(X2) mark#(fst(X1,X2)) = 2X1 + 1X2 >= X1 = mark#(X1) mark#(fst(X1,X2)) = 2X1 + 1X2 >= 1X2 = a__fst#(mark(X1),mark(X2)) a__fst#(s(X),cons(Y,Z)) = 1Y + 1Z >= Y = mark#(Y) mark#(from(X)) = X + 1 >= X = mark#(X) mark#(from(X)) = X + 1 >= X + 0 = a__from#(mark(X)) a__from#(X) = X + 0 >= X = mark#(X) mark#(add(X1,X2)) = X1 + 2X2 >= X2 = mark#(X2) mark#(add(X1,X2)) = X1 + 2X2 >= X1 = mark#(X1) mark#(add(X1,X2)) = X1 + 2X2 >= X1 + X2 = a__add#(mark(X1),mark(X2)) mark#(len(X)) = X + 0 >= X = mark#(X) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) a__fst(0(),Z) = 1Z + 2 >= 0 = nil() a__fst(s(X),cons(Y,Z)) = 2X + 1Y + 1Z + 2 >= 2X + Y + 1Z = cons(mark(Y),fst(X,Z)) a__from(X) = X + 1 >= X + 1 = cons(mark(X),from(s(X))) a__add(0(),X) = 2X + 0 >= X = mark(X) a__add(s(X),Y) = X + 2Y + 0 >= X + 2Y + 0 = s(add(X,Y)) a__len(nil()) = 0 >= 0 = 0() a__len(cons(X,Z)) = X + Z + 0 >= Z + 0 = s(len(Z)) mark(fst(X1,X2)) = 2X1 + 1X2 >= 2X1 + 1X2 = a__fst(mark(X1),mark(X2)) mark(from(X)) = X + 1 >= X + 1 = a__from(mark(X)) mark(add(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = a__add(mark(X1),mark(X2)) mark(len(X)) = X + 0 >= X + 0 = a__len(mark(X)) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(X) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) a__fst(X1,X2) = 2X1 + 1X2 >= 2X1 + 1X2 = fst(X1,X2) a__from(X) = X + 1 >= X + 1 = from(X) a__add(X1,X2) = X1 + 2X2 >= X1 + 2X2 = add(X1,X2) a__len(X) = X + 0 >= X + 0 = len(X) problem: DPs: a__add#(0(),X) -> mark#(X) mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(len(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Restore Modifier: DPs: a__add#(0(),X) -> mark#(X) mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(len(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) SCC Processor: #sccs: 1 #rules: 8 #arcs: 103/81 DPs: a__add#(0(),X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(len(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Arctic Interpretation Processor: dimension: 1 usable rules: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) interpretation: [a__add#](x0, x1) = x0 + 7x1 + 0, [a__from#](x0) = 5x0 + 0, [mark#](x0) = x0 + 0, [len](x0) = 1x0 + 2, [a__len](x0) = 1x0 + 2, [add](x0, x1) = 1x0 + 7x1, [a__add](x0, x1) = 1x0 + 7x1, [from](x0) = 5x0 + 0, [a__from](x0) = 5x0 + 0, [fst](x0, x1) = 2x0 + x1, [mark](x0) = x0, [cons](x0, x1) = x0, [s](x0) = 0, [nil] = 1, [a__fst](x0, x1) = 2x0 + x1, [0] = 0 orientation: a__add#(0(),X) = 7X + 0 >= X + 0 = mark#(X) mark#(from(X)) = 5X + 0 >= X + 0 = mark#(X) mark#(from(X)) = 5X + 0 >= 5X + 0 = a__from#(mark(X)) a__from#(X) = 5X + 0 >= X + 0 = mark#(X) mark#(add(X1,X2)) = 1X1 + 7X2 + 0 >= X1 + 0 = mark#(X1) mark#(add(X1,X2)) = 1X1 + 7X2 + 0 >= X1 + 7X2 + 0 = a__add#(mark(X1),mark(X2)) mark#(len(X)) = 1X + 2 >= X + 0 = mark#(X) mark#(cons(X1,X2)) = X1 + 0 >= X1 + 0 = mark#(X1) a__fst(0(),Z) = Z + 2 >= 1 = nil() a__fst(s(X),cons(Y,Z)) = Y + 2 >= Y = cons(mark(Y),fst(X,Z)) a__from(X) = 5X + 0 >= X = cons(mark(X),from(s(X))) a__add(0(),X) = 7X + 1 >= X = mark(X) a__add(s(X),Y) = 7Y + 1 >= 0 = s(add(X,Y)) a__len(nil()) = 2 >= 0 = 0() a__len(cons(X,Z)) = 1X + 2 >= 0 = s(len(Z)) mark(fst(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__fst(mark(X1),mark(X2)) mark(from(X)) = 5X + 0 >= 5X + 0 = a__from(mark(X)) mark(add(X1,X2)) = 1X1 + 7X2 >= 1X1 + 7X2 = a__add(mark(X1),mark(X2)) mark(len(X)) = 1X + 2 >= 1X + 2 = a__len(mark(X)) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 0 >= 0 = s(X) mark(nil()) = 1 >= 1 = nil() mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) a__fst(X1,X2) = 2X1 + X2 >= 2X1 + X2 = fst(X1,X2) a__from(X) = 5X + 0 >= 5X + 0 = from(X) a__add(X1,X2) = 1X1 + 7X2 >= 1X1 + 7X2 = add(X1,X2) a__len(X) = 1X + 2 >= 1X + 2 = len(X) problem: DPs: a__add#(0(),X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Restore Modifier: DPs: a__add#(0(),X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Arctic Interpretation Processor: dimension: 1 usable rules: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) interpretation: [a__add#](x0, x1) = -2x0 + x1 + -16, [a__from#](x0) = x0, [mark#](x0) = x0, [len](x0) = x0, [a__len](x0) = x0, [add](x0, x1) = 7x0 + x1 + 7, [a__add](x0, x1) = 7x0 + x1 + 7, [from](x0) = x0, [a__from](x0) = x0, [fst](x0, x1) = x0 + x1, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [s](x0) = x0, [nil] = 0, [a__fst](x0, x1) = x0 + x1, [0] = 0 orientation: a__add#(0(),X) = X + -2 >= X = mark#(X) mark#(from(X)) = X >= X = mark#(X) mark#(from(X)) = X >= X = a__from#(mark(X)) a__from#(X) = X >= X = mark#(X) mark#(add(X1,X2)) = 7X1 + X2 + 7 >= X1 = mark#(X1) mark#(add(X1,X2)) = 7X1 + X2 + 7 >= -2X1 + X2 + -16 = a__add#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) a__fst(0(),Z) = Z + 0 >= 0 = nil() a__fst(s(X),cons(Y,Z)) = X + Y + Z >= X + Y + Z = cons(mark(Y),fst(X,Z)) a__from(X) = X >= X = cons(mark(X),from(s(X))) a__add(0(),X) = X + 7 >= X = mark(X) a__add(s(X),Y) = 7X + Y + 7 >= 7X + Y + 7 = s(add(X,Y)) a__len(nil()) = 0 >= 0 = 0() a__len(cons(X,Z)) = X + Z >= Z = s(len(Z)) mark(fst(X1,X2)) = X1 + X2 >= X1 + X2 = a__fst(mark(X1),mark(X2)) mark(from(X)) = X >= X = a__from(mark(X)) mark(add(X1,X2)) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = a__add(mark(X1),mark(X2)) mark(len(X)) = X >= X = a__len(mark(X)) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(X) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) a__fst(X1,X2) = X1 + X2 >= X1 + X2 = fst(X1,X2) a__from(X) = X >= X = from(X) a__add(X1,X2) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = add(X1,X2) a__len(X) = X >= X = len(X) problem: DPs: a__add#(0(),X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Restore Modifier: DPs: a__add#(0(),X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Matrix Interpretation Processor: dim=1 usable rules: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) interpretation: [a__add#](x0, x1) = 4x1 + 2, [a__from#](x0) = 4x0, [mark#](x0) = 4x0, [len](x0) = 0, [a__len](x0) = 4, [add](x0, x1) = 4x1 + 6, [a__add](x0, x1) = 4x1 + 7, [from](x0) = 4x0 + 4, [a__from](x0) = 4x0 + 4, [fst](x0, x1) = x0 + 4x1 + 6, [mark](x0) = 4x0 + 4, [cons](x0, x1) = x0, [s](x0) = 0, [nil] = 0, [a__fst](x0, x1) = x0 + 4x1 + 6, [0] = 3 orientation: a__add#(0(),X) = 4X + 2 >= 4X = mark#(X) mark#(from(X)) = 16X + 16 >= 4X = mark#(X) mark#(from(X)) = 16X + 16 >= 16X + 16 = a__from#(mark(X)) a__from#(X) = 4X >= 4X = mark#(X) mark#(add(X1,X2)) = 16X2 + 24 >= 16X2 + 18 = a__add#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = 4X1 >= 4X1 = mark#(X1) a__fst(0(),Z) = 4Z + 9 >= 0 = nil() a__fst(s(X),cons(Y,Z)) = 4Y + 6 >= 4Y + 4 = cons(mark(Y),fst(X,Z)) a__from(X) = 4X + 4 >= 4X + 4 = cons(mark(X),from(s(X))) a__add(0(),X) = 4X + 7 >= 4X + 4 = mark(X) a__add(s(X),Y) = 4Y + 7 >= 0 = s(add(X,Y)) a__len(nil()) = 4 >= 3 = 0() a__len(cons(X,Z)) = 4 >= 0 = s(len(Z)) mark(fst(X1,X2)) = 4X1 + 16X2 + 28 >= 4X1 + 16X2 + 26 = a__fst(mark(X1),mark(X2)) mark(from(X)) = 16X + 20 >= 16X + 20 = a__from(mark(X)) mark(add(X1,X2)) = 16X2 + 28 >= 16X2 + 23 = a__add(mark(X1),mark(X2)) mark(len(X)) = 4 >= 4 = a__len(mark(X)) mark(0()) = 16 >= 3 = 0() mark(s(X)) = 4 >= 0 = s(X) mark(nil()) = 4 >= 0 = nil() mark(cons(X1,X2)) = 4X1 + 4 >= 4X1 + 4 = cons(mark(X1),X2) a__fst(X1,X2) = X1 + 4X2 + 6 >= X1 + 4X2 + 6 = fst(X1,X2) a__from(X) = 4X + 4 >= 4X + 4 = from(X) a__add(X1,X2) = 4X2 + 7 >= 4X2 + 6 = add(X1,X2) a__len(X) = 4 >= 0 = len(X) problem: DPs: mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Restore Modifier: DPs: mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Arctic Interpretation Processor: dimension: 1 usable rules: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) interpretation: [a__from#](x0) = 1x0 + 0, [mark#](x0) = 1x0 + 0, [len](x0) = 2x0 + 2, [a__len](x0) = 2x0 + 2, [add](x0, x1) = 3x0 + 2x1 + 3, [a__add](x0, x1) = 3x0 + 2x1 + 3, [from](x0) = 1x0 + 1, [a__from](x0) = 1x0 + 1, [fst](x0, x1) = x0 + x1 + 0, [mark](x0) = x0 + 0, [cons](x0, x1) = x0 + x1, [s](x0) = x0 + 0, [nil] = 2, [a__fst](x0, x1) = x0 + x1 + 0, [0] = 2 orientation: mark#(from(X)) = 2X + 2 >= 1X + 1 = a__from#(mark(X)) a__from#(X) = 1X + 0 >= 1X + 0 = mark#(X) mark#(cons(X1,X2)) = 1X1 + 1X2 + 0 >= 1X1 + 0 = mark#(X1) a__fst(0(),Z) = Z + 2 >= 2 = nil() a__fst(s(X),cons(Y,Z)) = X + Y + Z + 0 >= X + Y + Z + 0 = cons(mark(Y),fst(X,Z)) a__from(X) = 1X + 1 >= 1X + 1 = cons(mark(X),from(s(X))) a__add(0(),X) = 2X + 5 >= X + 0 = mark(X) a__add(s(X),Y) = 3X + 2Y + 3 >= 3X + 2Y + 3 = s(add(X,Y)) a__len(nil()) = 4 >= 2 = 0() a__len(cons(X,Z)) = 2X + 2Z + 2 >= 2Z + 2 = s(len(Z)) mark(fst(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__fst(mark(X1),mark(X2)) mark(from(X)) = 1X + 1 >= 1X + 1 = a__from(mark(X)) mark(add(X1,X2)) = 3X1 + 2X2 + 3 >= 3X1 + 2X2 + 3 = a__add(mark(X1),mark(X2)) mark(len(X)) = 2X + 2 >= 2X + 2 = a__len(mark(X)) mark(0()) = 2 >= 2 = 0() mark(s(X)) = X + 0 >= X + 0 = s(X) mark(nil()) = 2 >= 2 = nil() mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) a__fst(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = fst(X1,X2) a__from(X) = 1X + 1 >= 1X + 1 = from(X) a__add(X1,X2) = 3X1 + 2X2 + 3 >= 3X1 + 2X2 + 3 = add(X1,X2) a__len(X) = 2X + 2 >= 2X + 2 = len(X) problem: DPs: a__from#(X) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Restore Modifier: DPs: a__from#(X) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) SCC Processor: #sccs: 1 #rules: 1 #arcs: 38/4 DPs: mark#(cons(X1,X2)) -> mark#(X1) TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Subterm Criterion Processor: simple projection: pi(mark#) = 0 problem: DPs: TRS: a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__from(X) -> cons(mark(X),from(s(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__len(nil()) -> 0() a__len(cons(X,Z)) -> s(len(Z)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__fst(X1,X2) -> fst(X1,X2) a__from(X) -> from(X) a__add(X1,X2) -> add(X1,X2) a__len(X) -> len(X) Qed