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) Matrix Interpretation Processor: dimension: 1 interpretation: [a__add#](x0, x1) = x1, [a__from#](x0) = x0, [mark#](x0) = x0, [a__fst#](x0, x1) = x1, [len](x0) = x0 + 1, [a__len](x0) = x0 + 1, [add](x0, x1) = x0 + x1, [a__add](x0, x1) = x0 + x1, [from](x0) = x0, [a__from](x0) = x0, [fst](x0, x1) = x0 + x1, [mark](x0) = x0, [cons](x0, x1) = x0, [s](x0) = 0, [nil] = 0, [a__fst](x0, x1) = x0 + x1, [0] = 0 orientation: a__add#(0(),X) = X >= X = mark#(X) mark#(fst(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(fst(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(fst(X1,X2)) = X1 + X2 >= X2 = a__fst#(mark(X1),mark(X2)) a__fst#(s(X),cons(Y,Z)) = Y >= Y = mark#(Y) 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)) = X1 + X2 >= X2 = mark#(X2) mark#(add(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(add(X1,X2)) = X1 + X2 >= X2 = a__add#(mark(X1),mark(X2)) mark#(len(X)) = X + 1 >= X = mark#(X) mark#(cons(X1,X2)) = X1 >= X1 = mark#(X1) a__fst(0(),Z) = Z >= 0 = nil() a__fst(s(X),cons(Y,Z)) = Y >= Y = cons(mark(Y),fst(X,Z)) a__from(X) = X >= X = cons(mark(X),from(s(X))) a__add(0(),X) = X >= X = mark(X) a__add(s(X),Y) = Y >= 0 = s(add(X,Y)) a__len(nil()) = 1 >= 0 = 0() a__len(cons(X,Z)) = X + 1 >= 0 = 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)) = X1 + X2 >= X1 + X2 = a__add(mark(X1),mark(X2)) mark(len(X)) = X + 1 >= X + 1 = a__len(mark(X)) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 0 >= 0 = s(X) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 >= X1 = 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) = X1 + X2 >= X1 + X2 = add(X1,X2) a__len(X) = X + 1 >= X + 1 = len(X) problem: 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#(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: dimension: 1 interpretation: [a__add#](x0, x1) = x0 + x1 + 1, [a__from#](x0) = x0 + 1, [mark#](x0) = x0 + 1, [a__fst#](x0, x1) = x1 + 1, [len](x0) = 1, [a__len](x0) = 1, [add](x0, x1) = x0 + x1, [a__add](x0, x1) = x0 + x1, [from](x0) = x0 + 1, [a__from](x0) = x0 + 1, [fst](x0, x1) = x0 + x1, [mark](x0) = x0, [cons](x0, x1) = x0 + 1, [s](x0) = 0, [nil] = 0, [a__fst](x0, x1) = x0 + x1, [0] = 1 orientation: a__add#(0(),X) = X + 2 >= X + 1 = mark#(X) mark#(fst(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) mark#(fst(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(fst(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = a__fst#(mark(X1),mark(X2)) a__fst#(s(X),cons(Y,Z)) = Y + 2 >= Y + 1 = mark#(Y) mark#(from(X)) = X + 2 >= X + 1 = mark#(X) mark#(from(X)) = X + 2 >= X + 1 = a__from#(mark(X)) a__from#(X) = X + 1 >= X + 1 = mark#(X) mark#(add(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) mark#(add(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__add#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = mark#(X1) a__fst(0(),Z) = Z + 1 >= 0 = nil() a__fst(s(X),cons(Y,Z)) = Y + 1 >= Y + 1 = cons(mark(Y),fst(X,Z)) a__from(X) = X + 1 >= X + 1 = cons(mark(X),from(s(X))) a__add(0(),X) = X + 1 >= X = mark(X) a__add(s(X),Y) = Y >= 0 = s(add(X,Y)) a__len(nil()) = 1 >= 1 = 0() a__len(cons(X,Z)) = 1 >= 0 = s(len(Z)) mark(fst(X1,X2)) = X1 + X2 >= X1 + X2 = a__fst(mark(X1),mark(X2)) mark(from(X)) = X + 1 >= X + 1 = a__from(mark(X)) mark(add(X1,X2)) = X1 + X2 >= X1 + X2 = a__add(mark(X1),mark(X2)) mark(len(X)) = 1 >= 1 = a__len(mark(X)) mark(0()) = 1 >= 1 = 0() mark(s(X)) = 0 >= 0 = s(X) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(mark(X1),X2) a__fst(X1,X2) = X1 + X2 >= X1 + X2 = fst(X1,X2) a__from(X) = X + 1 >= X + 1 = from(X) a__add(X1,X2) = X1 + X2 >= X1 + X2 = add(X1,X2) a__len(X) = 1 >= 1 = len(X) problem: DPs: mark#(fst(X1,X2)) -> mark#(X2) mark#(fst(X1,X2)) -> mark#(X1) mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2)) 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)) 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: dimension: 1 interpretation: [a__add#](x0, x1) = x1, [a__from#](x0) = x0 + 1, [mark#](x0) = x0 + 1, [a__fst#](x0, x1) = 0, [len](x0) = 0, [a__len](x0) = 0, [add](x0, x1) = x0 + x1, [a__add](x0, x1) = x0 + x1, [from](x0) = 0, [a__from](x0) = 0, [fst](x0, x1) = x0 + x1 + 1, [mark](x0) = x0, [cons](x0, x1) = 0, [s](x0) = 0, [nil] = 0, [a__fst](x0, x1) = x0 + x1 + 1, [0] = 0 orientation: mark#(fst(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = mark#(X2) mark#(fst(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = mark#(X1) mark#(fst(X1,X2)) = X1 + X2 + 2 >= 0 = a__fst#(mark(X1),mark(X2)) a__from#(X) = X + 1 >= X + 1 = mark#(X) mark#(add(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) mark#(add(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(add(X1,X2)) = X1 + X2 + 1 >= X2 = a__add#(mark(X1),mark(X2)) a__fst(0(),Z) = Z + 1 >= 0 = nil() a__fst(s(X),cons(Y,Z)) = 1 >= 0 = cons(mark(Y),fst(X,Z)) a__from(X) = 0 >= 0 = cons(mark(X),from(s(X))) a__add(0(),X) = X >= X = mark(X) a__add(s(X),Y) = Y >= 0 = s(add(X,Y)) a__len(nil()) = 0 >= 0 = 0() a__len(cons(X,Z)) = 0 >= 0 = s(len(Z)) mark(fst(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__fst(mark(X1),mark(X2)) mark(from(X)) = 0 >= 0 = a__from(mark(X)) mark(add(X1,X2)) = X1 + X2 >= X1 + X2 = a__add(mark(X1),mark(X2)) mark(len(X)) = 0 >= 0 = a__len(mark(X)) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 0 >= 0 = s(X) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) a__fst(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = fst(X1,X2) a__from(X) = 0 >= 0 = from(X) a__add(X1,X2) = X1 + X2 >= X1 + X2 = add(X1,X2) a__len(X) = 0 >= 0 = len(X) problem: DPs: a__from#(X) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) mark#(add(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: dimension: 1 interpretation: [a__from#](x0) = 1, [mark#](x0) = 0, [len](x0) = 0, [a__len](x0) = 0, [add](x0, x1) = 0, [a__add](x0, x1) = 0, [from](x0) = 0, [a__from](x0) = 0, [fst](x0, x1) = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [s](x0) = 0, [nil] = 0, [a__fst](x0, x1) = 0, [0] = 0 orientation: a__from#(X) = 1 >= 0 = mark#(X) mark#(add(X1,X2)) = 0 >= 0 = mark#(X2) mark#(add(X1,X2)) = 0 >= 0 = mark#(X1) a__fst(0(),Z) = 0 >= 0 = nil() a__fst(s(X),cons(Y,Z)) = 0 >= 0 = cons(mark(Y),fst(X,Z)) a__from(X) = 0 >= 0 = cons(mark(X),from(s(X))) a__add(0(),X) = 0 >= 0 = mark(X) a__add(s(X),Y) = 0 >= 0 = s(add(X,Y)) a__len(nil()) = 0 >= 0 = 0() a__len(cons(X,Z)) = 0 >= 0 = s(len(Z)) mark(fst(X1,X2)) = 0 >= 0 = a__fst(mark(X1),mark(X2)) mark(from(X)) = 0 >= 0 = a__from(mark(X)) mark(add(X1,X2)) = 0 >= 0 = a__add(mark(X1),mark(X2)) mark(len(X)) = 0 >= 0 = a__len(mark(X)) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 0 >= 0 = s(X) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) a__fst(X1,X2) = 0 >= 0 = fst(X1,X2) a__from(X) = 0 >= 0 = from(X) a__add(X1,X2) = 0 >= 0 = add(X1,X2) a__len(X) = 0 >= 0 = len(X) problem: DPs: mark#(add(X1,X2)) -> mark#(X2) mark#(add(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: dimension: 1 interpretation: [mark#](x0) = x0 + 1, [len](x0) = 1, [a__len](x0) = 1, [add](x0, x1) = x0 + x1 + 1, [a__add](x0, x1) = x0 + x1 + 1, [from](x0) = 0, [a__from](x0) = 0, [fst](x0, x1) = x0 + 1, [mark](x0) = x0, [cons](x0, x1) = 0, [s](x0) = 1, [nil] = 1, [a__fst](x0, x1) = x0 + 1, [0] = 1 orientation: mark#(add(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = mark#(X2) mark#(add(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = mark#(X1) a__fst(0(),Z) = 2 >= 1 = nil() a__fst(s(X),cons(Y,Z)) = 2 >= 0 = cons(mark(Y),fst(X,Z)) a__from(X) = 0 >= 0 = cons(mark(X),from(s(X))) a__add(0(),X) = X + 2 >= X = mark(X) a__add(s(X),Y) = Y + 2 >= 1 = s(add(X,Y)) a__len(nil()) = 1 >= 1 = 0() a__len(cons(X,Z)) = 1 >= 1 = s(len(Z)) mark(fst(X1,X2)) = X1 + 1 >= X1 + 1 = a__fst(mark(X1),mark(X2)) mark(from(X)) = 0 >= 0 = a__from(mark(X)) mark(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__add(mark(X1),mark(X2)) mark(len(X)) = 1 >= 1 = a__len(mark(X)) mark(0()) = 1 >= 1 = 0() mark(s(X)) = 1 >= 1 = s(X) mark(nil()) = 1 >= 1 = nil() mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) a__fst(X1,X2) = X1 + 1 >= X1 + 1 = fst(X1,X2) a__from(X) = 0 >= 0 = from(X) a__add(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = add(X1,X2) a__len(X) = 1 >= 1 = len(X) 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