YES Problem: fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) from(X) -> cons(X,n__from(n__s(X))) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) len(nil()) -> 0() len(cons(X,Z)) -> s(n__len(activate(Z))) fst(X1,X2) -> n__fst(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) add(X1,X2) -> n__add(X1,X2) len(X) -> n__len(X) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(X) -> X Proof: DP Processor: DPs: fst#(s(X),cons(Y,Z)) -> activate#(Z) fst#(s(X),cons(Y,Z)) -> activate#(X) add#(s(X),Y) -> activate#(X) add#(s(X),Y) -> s#(n__add(activate(X),Y)) len#(cons(X,Z)) -> activate#(Z) len#(cons(X,Z)) -> s#(n__len(activate(Z))) activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) activate#(n__from(X)) -> activate#(X) activate#(n__from(X)) -> from#(activate(X)) activate#(n__s(X)) -> s#(X) activate#(n__add(X1,X2)) -> activate#(X2) activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) activate#(n__len(X)) -> activate#(X) activate#(n__len(X)) -> len#(activate(X)) TRS: fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) from(X) -> cons(X,n__from(n__s(X))) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) len(nil()) -> 0() len(cons(X,Z)) -> s(n__len(activate(Z))) fst(X1,X2) -> n__fst(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) add(X1,X2) -> n__add(X1,X2) len(X) -> n__len(X) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(X) -> X TDG Processor: DPs: fst#(s(X),cons(Y,Z)) -> activate#(Z) fst#(s(X),cons(Y,Z)) -> activate#(X) add#(s(X),Y) -> activate#(X) add#(s(X),Y) -> s#(n__add(activate(X),Y)) len#(cons(X,Z)) -> activate#(Z) len#(cons(X,Z)) -> s#(n__len(activate(Z))) activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) activate#(n__from(X)) -> activate#(X) activate#(n__from(X)) -> from#(activate(X)) activate#(n__s(X)) -> s#(X) activate#(n__add(X1,X2)) -> activate#(X2) activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) activate#(n__len(X)) -> activate#(X) activate#(n__len(X)) -> len#(activate(X)) TRS: fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) from(X) -> cons(X,n__from(n__s(X))) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) len(nil()) -> 0() len(cons(X,Z)) -> s(n__len(activate(Z))) fst(X1,X2) -> n__fst(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) add(X1,X2) -> n__add(X1,X2) len(X) -> n__len(X) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(X) -> X graph: len#(cons(X,Z)) -> activate#(Z) -> activate#(n__len(X)) -> len#(activate(X)) len#(cons(X,Z)) -> activate#(Z) -> activate#(n__len(X)) -> activate#(X) len#(cons(X,Z)) -> activate#(Z) -> activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) len#(cons(X,Z)) -> activate#(Z) -> activate#(n__add(X1,X2)) -> activate#(X1) len#(cons(X,Z)) -> activate#(Z) -> activate#(n__add(X1,X2)) -> activate#(X2) len#(cons(X,Z)) -> activate#(Z) -> activate#(n__s(X)) -> s#(X) len#(cons(X,Z)) -> activate#(Z) -> activate#(n__from(X)) -> from#(activate(X)) len#(cons(X,Z)) -> activate#(Z) -> activate#(n__from(X)) -> activate#(X) len#(cons(X,Z)) -> activate#(Z) -> activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) len#(cons(X,Z)) -> activate#(Z) -> activate#(n__fst(X1,X2)) -> activate#(X1) len#(cons(X,Z)) -> activate#(Z) -> activate#(n__fst(X1,X2)) -> activate#(X2) add#(s(X),Y) -> activate#(X) -> activate#(n__len(X)) -> len#(activate(X)) add#(s(X),Y) -> activate#(X) -> activate#(n__len(X)) -> activate#(X) add#(s(X),Y) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) add#(s(X),Y) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X1) add#(s(X),Y) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X2) add#(s(X),Y) -> activate#(X) -> activate#(n__s(X)) -> s#(X) add#(s(X),Y) -> activate#(X) -> activate#(n__from(X)) -> from#(activate(X)) add#(s(X),Y) -> activate#(X) -> activate#(n__from(X)) -> activate#(X) add#(s(X),Y) -> activate#(X) -> activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) add#(s(X),Y) -> activate#(X) -> activate#(n__fst(X1,X2)) -> activate#(X1) add#(s(X),Y) -> activate#(X) -> activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__len(X)) -> len#(activate(X)) -> len#(cons(X,Z)) -> s#(n__len(activate(Z))) activate#(n__len(X)) -> len#(activate(X)) -> len#(cons(X,Z)) -> activate#(Z) activate#(n__len(X)) -> activate#(X) -> activate#(n__len(X)) -> len#(activate(X)) activate#(n__len(X)) -> activate#(X) -> activate#(n__len(X)) -> activate#(X) activate#(n__len(X)) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) activate#(n__len(X)) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__len(X)) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X2) activate#(n__len(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(X) activate#(n__len(X)) -> activate#(X) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__len(X)) -> activate#(X) -> activate#(n__from(X)) -> activate#(X) activate#(n__len(X)) -> activate#(X) -> activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) activate#(n__len(X)) -> activate#(X) -> activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__len(X)) -> activate#(X) -> activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) -> add#(s(X),Y) -> s#(n__add(activate(X),Y)) activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) -> add#(s(X),Y) -> activate#(X) activate#(n__add(X1,X2)) -> activate#(X2) -> activate#(n__len(X)) -> len#(activate(X)) activate#(n__add(X1,X2)) -> activate#(X2) -> activate#(n__len(X)) -> activate#(X) activate#(n__add(X1,X2)) -> activate#(X2) -> activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) activate#(n__add(X1,X2)) -> activate#(X2) -> activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> activate#(X2) -> activate#(n__add(X1,X2)) -> activate#(X2) activate#(n__add(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> s#(X) activate#(n__add(X1,X2)) -> activate#(X2) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__add(X1,X2)) -> activate#(X2) -> activate#(n__from(X)) -> activate#(X) activate#(n__add(X1,X2)) -> activate#(X2) -> activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) activate#(n__add(X1,X2)) -> activate#(X2) -> activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> activate#(X2) -> activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__len(X)) -> len#(activate(X)) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__len(X)) -> activate#(X) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__add(X1,X2)) -> activate#(X2) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(X) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> activate#(X) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> activate#(X1) -> activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__from(X)) -> activate#(X) -> activate#(n__len(X)) -> len#(activate(X)) activate#(n__from(X)) -> activate#(X) -> activate#(n__len(X)) -> activate#(X) activate#(n__from(X)) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) activate#(n__from(X)) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__from(X)) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X2) activate#(n__from(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(X) activate#(n__from(X)) -> activate#(X) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__from(X)) -> activate#(X) -> activate#(n__from(X)) -> activate#(X) activate#(n__from(X)) -> activate#(X) -> activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) activate#(n__from(X)) -> activate#(X) -> activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__from(X)) -> activate#(X) -> activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X2) -> activate#(n__len(X)) -> len#(activate(X)) activate#(n__fst(X1,X2)) -> activate#(X2) -> activate#(n__len(X)) -> activate#(X) activate#(n__fst(X1,X2)) -> activate#(X2) -> activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) activate#(n__fst(X1,X2)) -> activate#(X2) -> activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> activate#(X2) -> activate#(n__add(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> s#(X) activate#(n__fst(X1,X2)) -> activate#(X2) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__fst(X1,X2)) -> activate#(X2) -> activate#(n__from(X)) -> activate#(X) activate#(n__fst(X1,X2)) -> activate#(X2) -> activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) activate#(n__fst(X1,X2)) -> activate#(X2) -> activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> activate#(X2) -> activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X1) -> activate#(n__len(X)) -> len#(activate(X)) activate#(n__fst(X1,X2)) -> activate#(X1) -> activate#(n__len(X)) -> activate#(X) activate#(n__fst(X1,X2)) -> activate#(X1) -> activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) activate#(n__fst(X1,X2)) -> activate#(X1) -> activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> activate#(X1) -> activate#(n__add(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(X) activate#(n__fst(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__fst(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> activate#(X) activate#(n__fst(X1,X2)) -> activate#(X1) -> activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) activate#(n__fst(X1,X2)) -> activate#(X1) -> activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> activate#(X1) -> activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) -> fst#(s(X),cons(Y,Z)) -> activate#(X) activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) -> fst#(s(X),cons(Y,Z)) -> activate#(Z) fst#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__len(X)) -> len#(activate(X)) fst#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__len(X)) -> activate#(X) fst#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__add(X1,X2)) -> activate#(X1) fst#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__add(X1,X2)) -> activate#(X2) fst#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__s(X)) -> s#(X) fst#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__from(X)) -> from#(activate(X)) fst#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__from(X)) -> activate#(X) fst#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__fst(X1,X2)) -> activate#(X1) fst#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__fst(X1,X2)) -> activate#(X2) fst#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__len(X)) -> len#(activate(X)) fst#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__len(X)) -> activate#(X) fst#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X1) fst#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__add(X1,X2)) -> activate#(X2) fst#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__s(X)) -> s#(X) fst#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__from(X)) -> from#(activate(X)) fst#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__from(X)) -> activate#(X) fst#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__fst(X1,X2)) -> activate#(X1) fst#(s(X),cons(Y,Z)) -> activate#(X) -> activate#(n__fst(X1,X2)) -> activate#(X2) SCC Processor: #sccs: 1 #rules: 13 #arcs: 116/289 DPs: len#(cons(X,Z)) -> activate#(Z) activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__from(X)) -> activate#(X) activate#(n__add(X1,X2)) -> activate#(X2) activate#(n__add(X1,X2)) -> activate#(X1) activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) add#(s(X),Y) -> activate#(X) activate#(n__len(X)) -> activate#(X) activate#(n__len(X)) -> len#(activate(X)) fst#(s(X),cons(Y,Z)) -> activate#(X) TRS: fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) from(X) -> cons(X,n__from(n__s(X))) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) len(nil()) -> 0() len(cons(X,Z)) -> s(n__len(activate(Z))) fst(X1,X2) -> n__fst(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) add(X1,X2) -> n__add(X1,X2) len(X) -> n__len(X) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [len#](x0) = x0, [add#](x0, x1) = 4x0 + x1 + 6, [activate#](x0) = x0, [fst#](x0, x1) = x0 + x1, [n__len](x0) = x0, [len](x0) = x0, [n__add](x0, x1) = 6x0 + x1 + 6, [add](x0, x1) = 6x0 + x1 + 6, [n__from](x0) = x0 + 0, [n__s](x0) = x0 + 0, [from](x0) = x0 + 0, [n__fst](x0, x1) = x0 + x1 + 0, [activate](x0) = x0, [cons](x0, x1) = x0 + x1 + 0, [s](x0) = x0 + 0, [nil] = 0, [fst](x0, x1) = x0 + x1 + 0, [0] = 0 orientation: len#(cons(X,Z)) = X + Z + 0 >= Z = activate#(Z) activate#(n__fst(X1,X2)) = X1 + X2 + 0 >= X2 = activate#(X2) activate#(n__fst(X1,X2)) = X1 + X2 + 0 >= X1 = activate#(X1) activate#(n__fst(X1,X2)) = X1 + X2 + 0 >= X1 + X2 = fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= Z = activate#(Z) activate#(n__from(X)) = X + 0 >= X = activate#(X) activate#(n__add(X1,X2)) = 6X1 + X2 + 6 >= X2 = activate#(X2) activate#(n__add(X1,X2)) = 6X1 + X2 + 6 >= X1 = activate#(X1) activate#(n__add(X1,X2)) = 6X1 + X2 + 6 >= 4X1 + X2 + 6 = add#(activate(X1),activate(X2)) add#(s(X),Y) = 4X + Y + 6 >= X = activate#(X) activate#(n__len(X)) = X >= X = activate#(X) activate#(n__len(X)) = X >= X = len#(activate(X)) fst#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= X = activate#(X) fst(0(),Z) = Z + 0 >= 0 = nil() fst(s(X),cons(Y,Z)) = X + Y + Z + 0 >= X + Y + Z + 0 = cons(Y,n__fst(activate(X),activate(Z))) from(X) = X + 0 >= X + 0 = cons(X,n__from(n__s(X))) add(0(),X) = X + 6 >= X = X add(s(X),Y) = 6X + Y + 6 >= 6X + Y + 6 = s(n__add(activate(X),Y)) len(nil()) = 0 >= 0 = 0() len(cons(X,Z)) = X + Z + 0 >= Z + 0 = s(n__len(activate(Z))) fst(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = n__fst(X1,X2) from(X) = X + 0 >= X + 0 = n__from(X) s(X) = X + 0 >= X + 0 = n__s(X) add(X1,X2) = 6X1 + X2 + 6 >= 6X1 + X2 + 6 = n__add(X1,X2) len(X) = X >= X = n__len(X) activate(n__fst(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = fst(activate(X1),activate(X2)) activate(n__from(X)) = X + 0 >= X + 0 = from(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(X) activate(n__add(X1,X2)) = 6X1 + X2 + 6 >= 6X1 + X2 + 6 = add(activate(X1),activate(X2)) activate(n__len(X)) = X >= X = len(activate(X)) activate(X) = X >= X = X problem: DPs: len#(cons(X,Z)) -> activate#(Z) activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__from(X)) -> activate#(X) activate#(n__add(X1,X2)) -> activate#(X2) activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2)) activate#(n__len(X)) -> activate#(X) activate#(n__len(X)) -> len#(activate(X)) fst#(s(X),cons(Y,Z)) -> activate#(X) TRS: fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) from(X) -> cons(X,n__from(n__s(X))) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) len(nil()) -> 0() len(cons(X,Z)) -> s(n__len(activate(Z))) fst(X1,X2) -> n__fst(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) add(X1,X2) -> n__add(X1,X2) len(X) -> n__len(X) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(X) -> X SCC Processor: #sccs: 1 #rules: 10 #arcs: 94/121 DPs: len#(cons(X,Z)) -> activate#(Z) activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__from(X)) -> activate#(X) activate#(n__add(X1,X2)) -> activate#(X2) activate#(n__len(X)) -> activate#(X) activate#(n__len(X)) -> len#(activate(X)) fst#(s(X),cons(Y,Z)) -> activate#(X) TRS: fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) from(X) -> cons(X,n__from(n__s(X))) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) len(nil()) -> 0() len(cons(X,Z)) -> s(n__len(activate(Z))) fst(X1,X2) -> n__fst(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) add(X1,X2) -> n__add(X1,X2) len(X) -> n__len(X) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [len#](x0) = x0, [activate#](x0) = x0 + 0, [fst#](x0, x1) = 4x0 + x1, [n__len](x0) = x0 + 4, [len](x0) = x0 + 4, [n__add](x0, x1) = 3x1 + 1, [add](x0, x1) = 3x1 + 1, [n__from](x0) = x0 + 0, [n__s](x0) = x0 + 0, [from](x0) = x0 + 0, [n__fst](x0, x1) = 4x0 + x1, [activate](x0) = x0, [cons](x0, x1) = x0 + x1 + 0, [s](x0) = x0 + 0, [nil] = 5, [fst](x0, x1) = 4x0 + x1, [0] = 2 orientation: len#(cons(X,Z)) = X + Z + 0 >= Z + 0 = activate#(Z) activate#(n__fst(X1,X2)) = 4X1 + X2 + 0 >= X2 + 0 = activate#(X2) activate#(n__fst(X1,X2)) = 4X1 + X2 + 0 >= X1 + 0 = activate#(X1) activate#(n__fst(X1,X2)) = 4X1 + X2 + 0 >= 4X1 + X2 = fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) = 4X + Y + Z + 4 >= Z + 0 = activate#(Z) activate#(n__from(X)) = X + 0 >= X + 0 = activate#(X) activate#(n__add(X1,X2)) = 3X2 + 1 >= X2 + 0 = activate#(X2) activate#(n__len(X)) = X + 4 >= X + 0 = activate#(X) activate#(n__len(X)) = X + 4 >= X = len#(activate(X)) fst#(s(X),cons(Y,Z)) = 4X + Y + Z + 4 >= X + 0 = activate#(X) fst(0(),Z) = Z + 6 >= 5 = nil() fst(s(X),cons(Y,Z)) = 4X + Y + Z + 4 >= 4X + Y + Z + 0 = cons(Y,n__fst(activate(X),activate(Z))) from(X) = X + 0 >= X + 0 = cons(X,n__from(n__s(X))) add(0(),X) = 3X + 1 >= X = X add(s(X),Y) = 3Y + 1 >= 3Y + 1 = s(n__add(activate(X),Y)) len(nil()) = 5 >= 2 = 0() len(cons(X,Z)) = X + Z + 4 >= Z + 4 = s(n__len(activate(Z))) fst(X1,X2) = 4X1 + X2 >= 4X1 + X2 = n__fst(X1,X2) from(X) = X + 0 >= X + 0 = n__from(X) s(X) = X + 0 >= X + 0 = n__s(X) add(X1,X2) = 3X2 + 1 >= 3X2 + 1 = n__add(X1,X2) len(X) = X + 4 >= X + 4 = n__len(X) activate(n__fst(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = fst(activate(X1),activate(X2)) activate(n__from(X)) = X + 0 >= X + 0 = from(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(X) activate(n__add(X1,X2)) = 3X2 + 1 >= 3X2 + 1 = add(activate(X1),activate(X2)) activate(n__len(X)) = X + 4 >= X + 4 = len(activate(X)) activate(X) = X >= X = X problem: DPs: len#(cons(X,Z)) -> activate#(Z) activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__from(X)) -> activate#(X) activate#(n__len(X)) -> activate#(X) activate#(n__len(X)) -> len#(activate(X)) TRS: fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) from(X) -> cons(X,n__from(n__s(X))) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) len(nil()) -> 0() len(cons(X,Z)) -> s(n__len(activate(Z))) fst(X1,X2) -> n__fst(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) add(X1,X2) -> n__add(X1,X2) len(X) -> n__len(X) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [len#](x0) = 1x0 + 0, [activate#](x0) = x0 + 0, [fst#](x0, x1) = x0 + x1 + 0, [n__len](x0) = 1x0 + 1, [len](x0) = 1x0 + 1, [n__add](x0, x1) = 1x0 + x1 + 1, [add](x0, x1) = 1x0 + x1 + 1, [n__from](x0) = 1x0 + 1, [n__s](x0) = x0, [from](x0) = 1x0 + 1, [n__fst](x0, x1) = x0 + x1, [activate](x0) = x0 + 0, [cons](x0, x1) = x0 + x1, [s](x0) = x0, [nil] = 2, [fst](x0, x1) = x0 + x1 + 0, [0] = 2 orientation: len#(cons(X,Z)) = 1X + 1Z + 0 >= Z + 0 = activate#(Z) activate#(n__fst(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = activate#(X2) activate#(n__fst(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = activate#(X1) activate#(n__fst(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= Z + 0 = activate#(Z) activate#(n__from(X)) = 1X + 1 >= X + 0 = activate#(X) activate#(n__len(X)) = 1X + 1 >= X + 0 = activate#(X) activate#(n__len(X)) = 1X + 1 >= 1X + 1 = len#(activate(X)) fst(0(),Z) = Z + 2 >= 2 = nil() fst(s(X),cons(Y,Z)) = X + Y + Z + 0 >= X + Y + Z + 0 = cons(Y,n__fst(activate(X),activate(Z))) from(X) = 1X + 1 >= 1X + 1 = cons(X,n__from(n__s(X))) add(0(),X) = X + 3 >= X = X add(s(X),Y) = 1X + Y + 1 >= 1X + Y + 1 = s(n__add(activate(X),Y)) len(nil()) = 3 >= 2 = 0() len(cons(X,Z)) = 1X + 1Z + 1 >= 1Z + 1 = s(n__len(activate(Z))) fst(X1,X2) = X1 + X2 + 0 >= X1 + X2 = n__fst(X1,X2) from(X) = 1X + 1 >= 1X + 1 = n__from(X) s(X) = X >= X = n__s(X) add(X1,X2) = 1X1 + X2 + 1 >= 1X1 + X2 + 1 = n__add(X1,X2) len(X) = 1X + 1 >= 1X + 1 = n__len(X) activate(n__fst(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = fst(activate(X1),activate(X2)) activate(n__from(X)) = 1X + 1 >= 1X + 1 = from(activate(X)) activate(n__s(X)) = X + 0 >= X = s(X) activate(n__add(X1,X2)) = 1X1 + X2 + 1 >= 1X1 + X2 + 1 = add(activate(X1),activate(X2)) activate(n__len(X)) = 1X + 1 >= 1X + 1 = len(activate(X)) activate(X) = X + 0 >= X = X problem: DPs: len#(cons(X,Z)) -> activate#(Z) activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__len(X)) -> len#(activate(X)) TRS: fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) from(X) -> cons(X,n__from(n__s(X))) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) len(nil()) -> 0() len(cons(X,Z)) -> s(n__len(activate(Z))) fst(X1,X2) -> n__fst(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) add(X1,X2) -> n__add(X1,X2) len(X) -> n__len(X) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [len#](x0) = x0 + 0, [activate#](x0) = x0 + 0, [fst#](x0, x1) = x0 + x1 + 0, [n__len](x0) = 1x0 + 1, [len](x0) = 1x0 + 1, [n__add](x0, x1) = x0 + x1, [add](x0, x1) = x0 + x1 + 0, [n__from](x0) = 4x0 + 4, [n__s](x0) = x0, [from](x0) = 4x0 + 4, [n__fst](x0, x1) = x0 + x1, [activate](x0) = x0 + 0, [cons](x0, x1) = 4x0 + x1 + 0, [s](x0) = x0, [nil] = 0, [fst](x0, x1) = x0 + x1 + 0, [0] = 1 orientation: len#(cons(X,Z)) = 4X + Z + 0 >= Z + 0 = activate#(Z) activate#(n__fst(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = activate#(X2) activate#(n__fst(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = activate#(X1) activate#(n__fst(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) = X + 4Y + Z + 0 >= Z + 0 = activate#(Z) activate#(n__len(X)) = 1X + 1 >= X + 0 = len#(activate(X)) fst(0(),Z) = Z + 1 >= 0 = nil() fst(s(X),cons(Y,Z)) = X + 4Y + Z + 0 >= X + 4Y + Z + 0 = cons(Y,n__fst(activate(X),activate(Z))) from(X) = 4X + 4 >= 4X + 4 = cons(X,n__from(n__s(X))) add(0(),X) = X + 1 >= X = X add(s(X),Y) = X + Y + 0 >= X + Y + 0 = s(n__add(activate(X),Y)) len(nil()) = 1 >= 1 = 0() len(cons(X,Z)) = 5X + 1Z + 1 >= 1Z + 1 = s(n__len(activate(Z))) fst(X1,X2) = X1 + X2 + 0 >= X1 + X2 = n__fst(X1,X2) from(X) = 4X + 4 >= 4X + 4 = n__from(X) s(X) = X >= X = n__s(X) add(X1,X2) = X1 + X2 + 0 >= X1 + X2 = n__add(X1,X2) len(X) = 1X + 1 >= 1X + 1 = n__len(X) activate(n__fst(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = fst(activate(X1),activate(X2)) activate(n__from(X)) = 4X + 4 >= 4X + 4 = from(activate(X)) activate(n__s(X)) = X + 0 >= X = s(X) activate(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = add(activate(X1),activate(X2)) activate(n__len(X)) = 1X + 1 >= 1X + 1 = len(activate(X)) activate(X) = X + 0 >= X = X problem: DPs: len#(cons(X,Z)) -> activate#(Z) activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) -> activate#(Z) TRS: fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) from(X) -> cons(X,n__from(n__s(X))) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) len(nil()) -> 0() len(cons(X,Z)) -> s(n__len(activate(Z))) fst(X1,X2) -> n__fst(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) add(X1,X2) -> n__add(X1,X2) len(X) -> n__len(X) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(X) -> X SCC Processor: #sccs: 1 #rules: 4 #arcs: 59/25 DPs: activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> activate#(X1) activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) -> activate#(Z) TRS: fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) from(X) -> cons(X,n__from(n__s(X))) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) len(nil()) -> 0() len(cons(X,Z)) -> s(n__len(activate(Z))) fst(X1,X2) -> n__fst(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) add(X1,X2) -> n__add(X1,X2) len(X) -> n__len(X) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [activate#](x0) = x0, [fst#](x0, x1) = x1 + 0, [n__len](x0) = x0, [len](x0) = x0, [n__add](x0, x1) = x1 + 4, [add](x0, x1) = x1 + 4, [n__from](x0) = x0 + 6, [n__s](x0) = x0, [from](x0) = x0 + 6, [n__fst](x0, x1) = 7x0 + x1 + 0, [activate](x0) = x0, [cons](x0, x1) = x1 + 0, [s](x0) = x0, [nil] = 0, [fst](x0, x1) = 7x0 + x1 + 0, [0] = 0 orientation: activate#(n__fst(X1,X2)) = 7X1 + X2 + 0 >= X2 = activate#(X2) activate#(n__fst(X1,X2)) = 7X1 + X2 + 0 >= X1 = activate#(X1) activate#(n__fst(X1,X2)) = 7X1 + X2 + 0 >= X2 + 0 = fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) = Z + 0 >= Z = activate#(Z) fst(0(),Z) = Z + 7 >= 0 = nil() fst(s(X),cons(Y,Z)) = 7X + Z + 0 >= 7X + Z + 0 = cons(Y,n__fst(activate(X),activate(Z))) from(X) = X + 6 >= X + 6 = cons(X,n__from(n__s(X))) add(0(),X) = X + 4 >= X = X add(s(X),Y) = Y + 4 >= Y + 4 = s(n__add(activate(X),Y)) len(nil()) = 0 >= 0 = 0() len(cons(X,Z)) = Z + 0 >= Z = s(n__len(activate(Z))) fst(X1,X2) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = n__fst(X1,X2) from(X) = X + 6 >= X + 6 = n__from(X) s(X) = X >= X = n__s(X) add(X1,X2) = X2 + 4 >= X2 + 4 = n__add(X1,X2) len(X) = X >= X = n__len(X) activate(n__fst(X1,X2)) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = fst(activate(X1),activate(X2)) activate(n__from(X)) = X + 6 >= X + 6 = from(activate(X)) activate(n__s(X)) = X >= X = s(X) activate(n__add(X1,X2)) = X2 + 4 >= X2 + 4 = add(activate(X1),activate(X2)) activate(n__len(X)) = X >= X = len(activate(X)) activate(X) = X >= X = X problem: DPs: activate#(n__fst(X1,X2)) -> activate#(X2) activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) -> activate#(Z) TRS: fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) from(X) -> cons(X,n__from(n__s(X))) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) len(nil()) -> 0() len(cons(X,Z)) -> s(n__len(activate(Z))) fst(X1,X2) -> n__fst(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) add(X1,X2) -> n__add(X1,X2) len(X) -> n__len(X) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [activate#](x0) = x0, [fst#](x0, x1) = x0 + x1, [n__len](x0) = 4x0, [len](x0) = 4x0, [n__add](x0, x1) = 5x1 + 0, [add](x0, x1) = 5x1 + 0, [n__from](x0) = x0 + 0, [n__s](x0) = x0 + 0, [from](x0) = x0 + 0, [n__fst](x0, x1) = 2x0 + 4x1, [activate](x0) = x0, [cons](x0, x1) = x0 + x1 + 0, [s](x0) = x0 + 0, [nil] = 2, [fst](x0, x1) = 2x0 + 4x1, [0] = 3 orientation: activate#(n__fst(X1,X2)) = 2X1 + 4X2 >= X2 = activate#(X2) activate#(n__fst(X1,X2)) = 2X1 + 4X2 >= X1 + X2 = fst#(activate(X1),activate(X2)) fst#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= Z = activate#(Z) fst(0(),Z) = 4Z + 5 >= 2 = nil() fst(s(X),cons(Y,Z)) = 2X + 4Y + 4Z + 4 >= 2X + Y + 4Z + 0 = cons(Y,n__fst(activate(X),activate(Z))) from(X) = X + 0 >= X + 0 = cons(X,n__from(n__s(X))) add(0(),X) = 5X + 0 >= X = X add(s(X),Y) = 5Y + 0 >= 5Y + 0 = s(n__add(activate(X),Y)) len(nil()) = 6 >= 3 = 0() len(cons(X,Z)) = 4X + 4Z + 4 >= 4Z + 0 = s(n__len(activate(Z))) fst(X1,X2) = 2X1 + 4X2 >= 2X1 + 4X2 = n__fst(X1,X2) from(X) = X + 0 >= X + 0 = n__from(X) s(X) = X + 0 >= X + 0 = n__s(X) add(X1,X2) = 5X2 + 0 >= 5X2 + 0 = n__add(X1,X2) len(X) = 4X >= 4X = n__len(X) activate(n__fst(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = fst(activate(X1),activate(X2)) activate(n__from(X)) = X + 0 >= X + 0 = from(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(X) activate(n__add(X1,X2)) = 5X2 + 0 >= 5X2 + 0 = add(activate(X1),activate(X2)) activate(n__len(X)) = 4X >= 4X = len(activate(X)) activate(X) = X >= X = X problem: DPs: fst#(s(X),cons(Y,Z)) -> activate#(Z) TRS: fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) from(X) -> cons(X,n__from(n__s(X))) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) len(nil()) -> 0() len(cons(X,Z)) -> s(n__len(activate(Z))) fst(X1,X2) -> n__fst(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) add(X1,X2) -> n__add(X1,X2) len(X) -> n__len(X) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(X) -> X SCC Processor: #sccs: 0 #rules: 0 #arcs: 10/1